9). AB + CD = AAA, where AB and CD are two-digit numbers and AAA is a three digit number; A, B, C, and D are distinct positive integers. In the addition problem above, what is the value of C?
(A) 1
(B) 3
(C) 7
(D) 9
(E) Cannot be determined
Answer - D
Explanation -- AB + CD = AAA
AB and CD are two digit numbers, hence AAA must be 111
Therefore 1B + CD = 111
B can assume any value between 3 and 9
If B = 3, then CD = 111-13 = 98 and C = 9
If B = 9, then CD = 111-19 = 92 and C = 9
Hence for all values of B between 3 & 9 ---- C = 9
Thus the answer is D i.e (C = 9)
Thursday, September 28, 2006
Tuesday, September 05, 2006
Manhattan Challenge Problem of the week! - sept 4
The Power of Absolutes
If a and b are integers and a is not equal to b, is ab > 0?
(1) a^b > 0
(2) a^b is a non-zero integer
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
If a and b are integers and a is not equal to b, is ab > 0?
(1) a^b > 0
(2) a^b is a non-zero integer
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
Wednesday, August 30, 2006
Manhattan Challenge Problem of the week ! - Aug 28
High Functioning
The vertical position of an object can be approximated at any given time by the function: p(t) = rt – 5t2 + b where p(t) is the vertical position in meters, t is the time in seconds, and r and b are constants. After 2 seconds, the position of an object is 41 meters, and after 5 seconds the position is 26 meters. What is the position of the object, in meters, after 4 seconds?
(A) 24
(B) 26
(C) 39
(D) 41
(E) 45
Answer - D , for OE click on link below.
Manhattan challenge problem
The vertical position of an object can be approximated at any given time by the function: p(t) = rt – 5t2 + b where p(t) is the vertical position in meters, t is the time in seconds, and r and b are constants. After 2 seconds, the position of an object is 41 meters, and after 5 seconds the position is 26 meters. What is the position of the object, in meters, after 4 seconds?
(A) 24
(B) 26
(C) 39
(D) 41
(E) 45
Answer - D , for OE click on link below.
Manhattan challenge problem
Manhattan Challenge Problem of the week ! - Aug 7
Question
If j and k are positive integers where k > j, what is the value of the remainder when k is divided by j?
(1) There exists a positive integer m such that k = jm + 5.
(2) j > 5
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
For the answer click on the link below -
What remains to be seen ?
If j and k are positive integers where k > j, what is the value of the remainder when k is divided by j?
(1) There exists a positive integer m such that k = jm + 5.
(2) j > 5
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
For the answer click on the link below -
What remains to be seen ?
Manhattan Challenge Problem of the week ! - Aug 14
Question
A paint crew gets a rush order to paint 80 houses in a new development. They paint the first y houses at a rate of x houses per week. Realizing that they'll be late at this rate, they bring in some more painters and paint the rest of the houses at the rate of 1.25x houses per week. The total time it takes them to paint all the houses under this scenario is what fraction of the time it would have taken if they had painted all the houses at their original rate of x houses per week?
(A) 0.8(80 – y)
(B) 0.8 + 0.0025y
(C) 80/y – 1.25
(D) 80/1.25y
(E) 80 – 0.25y
For answer click on the link below -
Rush paint job
A paint crew gets a rush order to paint 80 houses in a new development. They paint the first y houses at a rate of x houses per week. Realizing that they'll be late at this rate, they bring in some more painters and paint the rest of the houses at the rate of 1.25x houses per week. The total time it takes them to paint all the houses under this scenario is what fraction of the time it would have taken if they had painted all the houses at their original rate of x houses per week?
(A) 0.8(80 – y)
(B) 0.8 + 0.0025y
(C) 80/y – 1.25
(D) 80/1.25y
(E) 80 – 0.25y
For answer click on the link below -
Rush paint job
Manhatten Challenge Problem of the week ! - Aug 21
Question
What is the ratio of 2x to 3y?
(1) The ratio of x^2 to y^2 is equal to 36/25.
(2) The ratio of x^5 to y^5 is greater than 1.
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
For the answer click on link below -
Ratios and exponents
What is the ratio of 2x to 3y?
(1) The ratio of x^2 to y^2 is equal to 36/25.
(2) The ratio of x^5 to y^5 is greater than 1.
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
For the answer click on link below -
Ratios and exponents
Monday, July 31, 2006
Manhattan Challenge Problem of the week ! july 24
Is the positive integer x odd?
(1) x = y2 + 4y + 6, where y is a positive integer.
(2) x = 9z2 + 7z - 10, where z is a positive integer.
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
Answer - The correct answer is B.
For complete solution click on the link below.
Manhattan challenge problem
(1) x = y2 + 4y + 6, where y is a positive integer.
(2) x = 9z2 + 7z - 10, where z is a positive integer.
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
Answer - The correct answer is B.
For complete solution click on the link below.
Manhattan challenge problem
Saturday, July 29, 2006
Maths - Prime Factors
Methods and tricks to solve questions related to Prime factors.
1). Counting the Number of Factors -- If you factor a number into its prime power factors, then the total number of factors is found by adding one to all the exponents and multiplying those results together.
Example: The total number of factors of 108 are --
108 = (2^2) * (3^3) thus we add 1 to the exponents and multiply the results together.
(2+1)*(3+1) = 3*4 = 12.
Hence total number of factors of 108 are 12.
Verifying the above
The factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108 , which are total 12 in number.
2). Factoring Factorials - First write the number you are factoring as a product of two or more numbers. For example, suppose we want to factor the number 30.
We know that 30 = 5 x 6,
so we write for the first step of our factor tree:
30
/ \
5 6
Next we factor the factors, if possible. In other words, for each number in the product from the first step, we try to write it down as a product of even smaller numbers.
For instance, in our example, we would try to factor both 5 and 6. As we noted above, 5 is prime, so we can't factor it further.
We are done with that branch of the factor tree.
However, 6 is not prime. 6 = 2 x 3, so we can extend our factor tree as follows:
30
/ \
5 6
Now further
6
/ \
2 3
Now, we continue this process until all of the branches of the tree end in prime numbers.
In our example, we are done after two steps, since 5, 2, and 3 are all prime numbers.
The factors of the number are the numbers at the end of the different branches on the factor tree. To figure out what power each prime factor is raised to, count the number of times the prime factor appears in the factor tree.
In our example, each of the factors appears only once, so the prime factorization of 30 is: 30 = 2 x 3 x 5.
1). Counting the Number of Factors -- If you factor a number into its prime power factors, then the total number of factors is found by adding one to all the exponents and multiplying those results together.
Example: The total number of factors of 108 are --
108 = (2^2) * (3^3) thus we add 1 to the exponents and multiply the results together.
(2+1)*(3+1) = 3*4 = 12.
Hence total number of factors of 108 are 12.
Verifying the above
The factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108 , which are total 12 in number.
2). Factoring Factorials - First write the number you are factoring as a product of two or more numbers. For example, suppose we want to factor the number 30.
We know that 30 = 5 x 6,
so we write for the first step of our factor tree:
30
/ \
5 6
Next we factor the factors, if possible. In other words, for each number in the product from the first step, we try to write it down as a product of even smaller numbers.
For instance, in our example, we would try to factor both 5 and 6. As we noted above, 5 is prime, so we can't factor it further.
We are done with that branch of the factor tree.
However, 6 is not prime. 6 = 2 x 3, so we can extend our factor tree as follows:
30
/ \
5 6
Now further
6
/ \
2 3
Now, we continue this process until all of the branches of the tree end in prime numbers.
In our example, we are done after two steps, since 5, 2, and 3 are all prime numbers.
The factors of the number are the numbers at the end of the different branches on the factor tree. To figure out what power each prime factor is raised to, count the number of times the prime factor appears in the factor tree.
In our example, each of the factors appears only once, so the prime factorization of 30 is: 30 = 2 x 3 x 5.
Thursday, July 27, 2006
Problem Solving - 8
How many factors of 2940 are NOT factors of 112?
(A) 22
(B) 30
(C) 32
(D) 34
(E) None of these
Answer - B
Prime factors of 2940 = (2 ^ 2) * 3 * 5 * (7 ^ 7)
Hence total number of factors of 2940
= (2 + 1) * ( 1 + 1) * (1 + 1) * ( 2 + 1) = 36
(by the method of counting number of factors)
Prime factors of 112 = (2 ^ 4) * 7
Therefore common factors for the above two numbers are
2^2 * 7
Total number of factors for (2 ^ 2) * 7 = (2 + 1) * ( 1 + 1) = 6
Hence total factors of 2940 which are not factors of 112 are
36 - 6 = 30, Hence B is the answer.
(A) 22
(B) 30
(C) 32
(D) 34
(E) None of these
Answer - B
Prime factors of 2940 = (2 ^ 2) * 3 * 5 * (7 ^ 7)
Hence total number of factors of 2940
= (2 + 1) * ( 1 + 1) * (1 + 1) * ( 2 + 1) = 36
(by the method of counting number of factors)
Prime factors of 112 = (2 ^ 4) * 7
Therefore common factors for the above two numbers are
2^2 * 7
Total number of factors for (2 ^ 2) * 7 = (2 + 1) * ( 1 + 1) = 6
Hence total factors of 2940 which are not factors of 112 are
36 - 6 = 30, Hence B is the answer.
Thursday, July 20, 2006
Problem Solving - 7
Eight points lie on the circumfrence of a circle.What is the positive diff b/w th no. of triangles and the no. of quadrilaterals that can be formed by connecting these points?
a. 8
b. 14
c. 56
d. 70
e. 1,344
Answer - B
Since the points are on the circumference of the circle thus no 3 or 4 points will be colinear, thus all points can be used to draw a triangle or quadilateral
Hence 8C4 = 70 and 8C3 = 56 (we make use of combination here because here the order of points do not matter)
Taking the difference we get 70 - 56 = 14.
a. 8
b. 14
c. 56
d. 70
e. 1,344
Answer - B
Since the points are on the circumference of the circle thus no 3 or 4 points will be colinear, thus all points can be used to draw a triangle or quadilateral
Hence 8C4 = 70 and 8C3 = 56 (we make use of combination here because here the order of points do not matter)
Taking the difference we get 70 - 56 = 14.
DS Question 13
Is the integer n odd?
(1) n is divisible by 3.
(2) 2n is divisible by twice as many positive integers as n.
Answer - B
Statement (1) - insufficient as n can be both even and odd.
Statement (2) - sufficient - Here n will have a unique prime factorisation as a product of
p1 ^ q1 * p2 ^ q2 * ....... pk ^qk
where p1 is less than p2 , p2 is less than p3 and so on.
Now p1 is either equal to 2 or it is an odd integer greater than 2.
Thus if n is odd then 2n has a unique prime factorisation of
2 * p1^q1 * p2^q2 * .... * pk ^ qk
but if n is even then 2n can be factorised as
p1^(q1+1) * p2^q2 * ... where p1=2, q1 >= 1
Hence , now the total number of factors is based on choice of prime powers for each prime in term.
Thus if n is odd, then 2n must have twice as many factors as n.
Alternatively in simpler language
If n is even --- 2n will not have twice as many divisors as n
e.g . For n=2; 2n or 4 has 1,2,4 as divisors ,
For n = 4 divisors are 1,2,4; 2n=8 has 1,2,4,8
But if n is odd
then if n=3 divisors are 1,3 and 2n=6 divisors are 1,2,3,6
n=5 divisors are 1,5 and 2n=10 divisors are 1,2,5,10
Thus if n is odd the 2nd condition is satisfied. Hence ans is B
(1) n is divisible by 3.
(2) 2n is divisible by twice as many positive integers as n.
Answer - B
Statement (1) - insufficient as n can be both even and odd.
Statement (2) - sufficient - Here n will have a unique prime factorisation as a product of
p1 ^ q1 * p2 ^ q2 * ....... pk ^qk
where p1 is less than p2 , p2 is less than p3 and so on.
Now p1 is either equal to 2 or it is an odd integer greater than 2.
Thus if n is odd then 2n has a unique prime factorisation of
2 * p1^q1 * p2^q2 * .... * pk ^ qk
but if n is even then 2n can be factorised as
p1^(q1+1) * p2^q2 * ... where p1=2, q1 >= 1
Hence , now the total number of factors is based on choice of prime powers for each prime in term.
Thus if n is odd, then 2n must have twice as many factors as n.
Alternatively in simpler language
If n is even --- 2n will not have twice as many divisors as n
e.g . For n=2; 2n or 4 has 1,2,4 as divisors ,
For n = 4 divisors are 1,2,4; 2n=8 has 1,2,4,8
But if n is odd
then if n=3 divisors are 1,3 and 2n=6 divisors are 1,2,3,6
n=5 divisors are 1,5 and 2n=10 divisors are 1,2,5,10
Thus if n is odd the 2nd condition is satisfied. Hence ans is B
Monday, July 17, 2006
Manhattan Challenge problem of the Week!
Dig those Digits - Challenge problem of the week
Click on the link below to see the question and its solution.
Manhattan challenge problem
Click on the link below to see the question and its solution.
Manhattan challenge problem
Wednesday, July 05, 2006
DS Question - 12
Does x + y = xy?
(1) x is neither a positive integer nor a negative integer
(2) y is neither a positive integer nor a negative integer
Answer - E - x and y can be either 0 or fractions.. which makes both conditions insufficient.
(1) x is neither a positive integer nor a negative integer
(2) y is neither a positive integer nor a negative integer
Answer - E - x and y can be either 0 or fractions.. which makes both conditions insufficient.
Manhattan Challenge Problem of the Week - July 5
Louie takes out a three-month loan of $1000. The lender charges him 10% interest per month compunded monthly. The terms of the loan state that Louie must repay the loan in three equal monthly payments. To the nearest dollar, how much does Louie have to pay each month?
(A) 333
(B) 383
(C) 402
(D) 433
(E) 483
Answer - C
For solution click on the link below
Manhattan challenge problem
(A) 333
(B) 383
(C) 402
(D) 433
(E) 483
Answer - C
For solution click on the link below
Manhattan challenge problem
Problem Solving - 6
For every positive integer n, the function h(n) is defined to be the product of all of the even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100) + 1, then p is
A. between 2 and 10
B. between 10 and 20
C. between 20 and 30
D. between 30 and 40
E. greater than 40
Answer - E
h(n)=2*4*6....100 =2(1*2*3*........50)
h(n) + 1 = 2(1*2*...50) + 1
Prime No ={2,3,........}
h(n) contains multiples of all the primes uptil 50 .... Hence E
OR
Let X = 2*4*6*.....*100
then X contains multiples of all the primes uptil 50
eg.
17 * 2 = 34 is in X.
37 * 2 = 74 is in X
Hence all the primes between 2 - 50 will be a factor of X => that none of them will be a factor of (X+1), hence the smallest prime should be at least greater than 50
Hence E.
A. between 2 and 10
B. between 10 and 20
C. between 20 and 30
D. between 30 and 40
E. greater than 40
Answer - E
h(n)=2*4*6....100 =2(1*2*3*........50)
h(n) + 1 = 2(1*2*...50) + 1
Prime No ={2,3,........}
h(n) contains multiples of all the primes uptil 50 .... Hence E
OR
Let X = 2*4*6*.....*100
then X contains multiples of all the primes uptil 50
eg.
17 * 2 = 34 is in X.
37 * 2 = 74 is in X
Hence all the primes between 2 - 50 will be a factor of X => that none of them will be a factor of (X+1), hence the smallest prime should be at least greater than 50
Hence E.
DS Question - 11
a, b, c all are positive, is a/b > (a+c)/(b+c) ?
1. a > c
2. a > b
Answer - B
statement (1). - insufficient as we do not know the relation between a and b
statement (2). - a > b gives answer irrespective of whether ( c <> b) that, a / b > (a + c) / (b + c)
a/b > [a+c]/[b+c]
= a(b+c) > b(a+c)
= ab+ac > ab+bc
= ac > bc
=a > b
1. a > c
2. a > b
Answer - B
statement (1). - insufficient as we do not know the relation between a and b
statement (2). - a > b gives answer irrespective of whether ( c <> b) that, a / b > (a + c) / (b + c)
a/b > [a+c]/[b+c]
= a(b+c) > b(a+c)
= ab+ac > ab+bc
= ac > bc
=a > b
Tuesday, July 04, 2006
DS Question - 10
When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15 ?
(1) n- 2 is divisible by 5.
(2) t is divisible by 3.
Answer - C
Statement (1) alone - no sufficient information about t
Statement (2)alone - no sufficient information about n
From the ques --- n = a*3 + 2
t = b*5 + 3
Using Statement (1) --- n = p*5 + 2
Thus n = 15*k + 2
Using Statement (2) --- t = q*3 + 3
Thus t = 15*j + 3
nt = 15*k(15*j + 3) + 30*j + 6
=> always leaves a remainder of 6 when divided by 15.
(1) n- 2 is divisible by 5.
(2) t is divisible by 3.
Answer - C
Statement (1) alone - no sufficient information about t
Statement (2)alone - no sufficient information about n
From the ques --- n = a*3 + 2
t = b*5 + 3
Using Statement (1) --- n = p*5 + 2
Thus n = 15*k + 2
Using Statement (2) --- t = q*3 + 3
Thus t = 15*j + 3
nt = 15*k(15*j + 3) + 30*j + 6
=> always leaves a remainder of 6 when divided by 15.
Problem Solving - 5
If |x| > 3, which of the following must be true?
A). x>3
B). x<3
C). x=3
D). x is not equal to 3
E). x<-3
Answer - D
A,B,E - Ruled out - a modulus sign in the question=>cannot have a single signed (can always find a solution with the opposite sign )
When x is positive ----- x >3
When x is negative ---- x <-3
Thus x can never be equal to 3.
A). x>3
B). x<3
C). x=3
D). x is not equal to 3
E). x<-3
Answer - D
A,B,E - Ruled out - a modulus sign in the question=>cannot have a single signed (can always find a solution with the opposite sign )
When x is positive ----- x >3
When x is negative ---- x <-3
Thus x can never be equal to 3.
DS Question - 9
Find an integer S.
(1) S=3
(2) S^3=3^S
Answer - B
Statement (1) - insufficient ---- S=3 or –3
Statement (2) - sufficient ---- S = 3
X^Y = Y^X only if X and Y are positive integers of equal value
=> X and Y have to be positive, otherwise one side will give a negative value when X or Y is odd while other side will give a decimal value
Given that Y is equal to 3, X has to be equal to 3.
(1) S=3
(2) S^3=3^S
Answer - B
Statement (1) - insufficient ---- S=3 or –3
Statement (2) - sufficient ---- S = 3
X^Y = Y^X only if X and Y are positive integers of equal value
=> X and Y have to be positive, otherwise one side will give a negative value when X or Y is odd while other side will give a decimal value
Given that Y is equal to 3, X has to be equal to 3.
Monday, July 03, 2006
Problem Solving - 4
What is the sum of all three digit positive integers, the sum of whose digits is a multiple of 3?
a) 900
b) 165150
c) 165000
d) 329199
e) 328900
Answer - B
Sum of n numbers = n/2(2a+(n-1)d)
We want to add the numbers from 102 to 999 that are divisible by 3.
We divide by 3 to get 34 to 333, or 300 numbers in total.
Here n = 300, a = 34, d= 1
Sum = 1/2*300*(333+34) = 150 * 367 = 55,050
Multiplying by 3 again gives 165,150 .
TIP
To add the first n integers from 1 to n
add together 1 + n = n+1
then 2 + (n-1) = n + 1 until you get to n/2 (depending on whether n is odd or even)
Ultimately we have n/2 pairs that all add up to (n+1)
So the sum is 1/2 * n * (n+1)
If you don't start at 1, or have a gap of more than 1 --- slightly adjust the formula.
a) 900
b) 165150
c) 165000
d) 329199
e) 328900
Answer - B
Sum of n numbers = n/2(2a+(n-1)d)
We want to add the numbers from 102 to 999 that are divisible by 3.
We divide by 3 to get 34 to 333, or 300 numbers in total.
Here n = 300, a = 34, d= 1
Sum = 1/2*300*(333+34) = 150 * 367 = 55,050
Multiplying by 3 again gives 165,150 .
TIP
To add the first n integers from 1 to n
add together 1 + n = n+1
then 2 + (n-1) = n + 1 until you get to n/2 (depending on whether n is odd or even)
Ultimately we have n/2 pairs that all add up to (n+1)
So the sum is 1/2 * n * (n+1)
If you don't start at 1, or have a gap of more than 1 --- slightly adjust the formula.
DS Question - 8
If X and Y are nonzero integers, what is the remainder when X is divided by Y?
(1) When X is divided by 2y, the remainder is 4
(2) When X+Y is divided by y, the remainder is 4
Answer - B
Statement (1) ---- x = (2y)M +4
if y > 4 the remainder of x/y will be 4
if y= 1 or 2 or 4 there will be no remainder,
if y=3, remainder will be 1,
so it depends on value of y...
so insufficient
Statement (2) -----(x+y)/y=M+4
It is clear that y/y=1 therefore (x/y) has to be 4 - Thus
sufficient
(1) When X is divided by 2y, the remainder is 4
(2) When X+Y is divided by y, the remainder is 4
Answer - B
Statement (1) ---- x = (2y)M +4
if y > 4 the remainder of x/y will be 4
if y= 1 or 2 or 4 there will be no remainder,
if y=3, remainder will be 1,
so it depends on value of y...
so insufficient
Statement (2) -----(x+y)/y=M+4
It is clear that y/y=1 therefore (x/y) has to be 4 - Thus
sufficient
Friday, June 30, 2006
Problem Solving - 3
There are two buses A and B. On Monday, bus A departs at 3pm and bus B at 4pm. After this, bus A departs every 10 hours and bus B every 15 hours. What is the earliest day they will depart at the same time?
A). Tuesday
B). Wednesday
C). Thursday
D). Sunday
E). so long as they continue operating in this manner, they will never leave at the same time.
Answer - E is the right choice.
The LCM of 10 hours and 15 hours = 30 hours,
30 hours = 6 hours mod 24 hour (day)
After that period they will just repeat the cycle.
A). Tuesday
B). Wednesday
C). Thursday
D). Sunday
E). so long as they continue operating in this manner, they will never leave at the same time.
Answer - E is the right choice.
The LCM of 10 hours and 15 hours = 30 hours,
30 hours = 6 hours mod 24 hour (day)
After that period they will just repeat the cycle.
Problem Solving - 2
If the positive integers m and n have the same two digits, but in reverse order, then the difference between m and n cannot be:
a) 24
b) 36
c) 54
d) 63
e) 72
Answer - (a)
Let m consists of digits x & y,
thus m = 10x+y and
n = 10y+x
m-n = 9x-9y = m-n = 9(x-y)
=> m-n has to be a multiple of 9....
a) 24
b) 36
c) 54
d) 63
e) 72
Answer - (a)
Let m consists of digits x & y,
thus m = 10x+y and
n = 10y+x
m-n = 9x-9y = m-n = 9(x-y)
=> m-n has to be a multiple of 9....
DS Question - 7
A building has two types of apartments, big and small. 65 percent of the apartments are small. The number of occupied big apartments is twice the number of unoccupied small apartments. What percent of the apartments in the building are occupied ?
(1) The number of occupied big apartments is six times the number of unoccupied big apartments.
(2) The building has a total of 160 apartments
Answer - A
Statement (1 ) -- Sufficient.
Percentage occupied = (big & occupied + small & occupied) / total (say, n)
(35/100)n * (6/7) = big & occupied
(small ) - (small & unoccupied) = (small & occupied)
(65/100)n - [(1/2) big & occupied] = small & occupied
(65/100)n - [(1/2) (35/100)n * (6/7)] = small & occupied
Total = [(35/100)n * (6/7) + (65/100)n - (35/100)n * (3/7) ] / n
(multiplied (6/7) by (1/2) here ).
n is removed from the numerator and divisor
(35/100) * (6/7) + (65/100) - (35/100) * (3/7) --- This is approx equal to 0.8
so whatever n is, it is the answer.
Statement (2) -- Insufficient -- 160 total apartments does not give any information regarding any number of occupied apartments -- there could be 2 big and 1 small, or 40 big and 20 small.
(1) The number of occupied big apartments is six times the number of unoccupied big apartments.
(2) The building has a total of 160 apartments
Answer - A
Statement (1 ) -- Sufficient.
Percentage occupied = (big & occupied + small & occupied) / total (say, n)
(35/100)n * (6/7) = big & occupied
(small ) - (small & unoccupied) = (small & occupied)
(65/100)n - [(1/2) big & occupied] = small & occupied
(65/100)n - [(1/2) (35/100)n * (6/7)] = small & occupied
Total = [(35/100)n * (6/7) + (65/100)n - (35/100)n * (3/7) ] / n
(multiplied (6/7) by (1/2) here ).
n is removed from the numerator and divisor
(35/100) * (6/7) + (65/100) - (35/100) * (3/7) --- This is approx equal to 0.8
so whatever n is, it is the answer.
Statement (2) -- Insufficient -- 160 total apartments does not give any information regarding any number of occupied apartments -- there could be 2 big and 1 small, or 40 big and 20 small.
Monday, June 26, 2006
Manhattan challenge problem of the week -- June 26
If set S = {7, y, 12, 8, x, 9}, is x + y less than 18?
(1) The range of set S is less than 9.
(2) The average of x and y is less than the average of set S.
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
Answer - B
(1) INSUFFICIENT: Statement (1) tells us that the range of S is less than 9. The range of a set is the positive difference between the smallest term and the largest term of the set. In this case, knowing that the range of set S is less than 9, we can answer only MAYBE to the question "Is (x + y) <>
Consider the following two examples:Let x = 7 and y = 7. The range of S is less than 9 and x + y <>
Let x = 10 and y = 10. The range of S is less than 9 and x + y > 18, so we conclude NO.
Because this statement does not allow us to answer definitively Yes or No, it is insufficient.
(2) SUFFICIENT: Statement (2) tells us that the average of x and y is less than the average of the set S. Writing this as an inequality:
(x + y)/2 < (7 + 8 + 9 + 12 + x + y)/6
(x + y)/2 < (36 + x + y)/6
3(x + y) <>
2(x + y) <>
x + y <>
Therefore, statement (2) is SUFFICIENT to determine whether x + y <>
(1) The range of set S is less than 9.
(2) The average of x and y is less than the average of set S.
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
Answer - B
(1) INSUFFICIENT: Statement (1) tells us that the range of S is less than 9. The range of a set is the positive difference between the smallest term and the largest term of the set. In this case, knowing that the range of set S is less than 9, we can answer only MAYBE to the question "Is (x + y) <>
Consider the following two examples:Let x = 7 and y = 7. The range of S is less than 9 and x + y <>
Let x = 10 and y = 10. The range of S is less than 9 and x + y > 18, so we conclude NO.
Because this statement does not allow us to answer definitively Yes or No, it is insufficient.
(2) SUFFICIENT: Statement (2) tells us that the average of x and y is less than the average of the set S. Writing this as an inequality:
(x + y)/2 < (7 + 8 + 9 + 12 + x + y)/6
(x + y)/2 < (36 + x + y)/6
3(x + y) <>
2(x + y) <>
x + y <>
Therefore, statement (2) is SUFFICIENT to determine whether x + y <>
Thursday, June 22, 2006
DS Question - 6
Give that n is an integer, is n-1 divisible by 3?
1. n^2 +n is not divisible by 3
2. 3n+5>=k+8, where k is a positive multiple of 3
Answer - A
Sttement (1) Sufficient -- It can be simplified as n(n+1) is not divisible by 3
i.e to satisfy the above condition n can take a value immediately following 3 or multiples of 3.
Such as n = 4, 7, 10 .....
Thus case n-1 is always divisible by 3.
Statement (2) Insufficient -- 3n >= k + 3
This implies n >= (k+3)/3
the possible values of k includes 3,6,9,12 .....
So this can be either multiple of 3 or any other number..
1. n^2 +n is not divisible by 3
2. 3n+5>=k+8, where k is a positive multiple of 3
Answer - A
Sttement (1) Sufficient -- It can be simplified as n(n+1) is not divisible by 3
i.e to satisfy the above condition n can take a value immediately following 3 or multiples of 3.
Such as n = 4, 7, 10 .....
Thus case n-1 is always divisible by 3.
Statement (2) Insufficient -- 3n >= k + 3
This implies n >= (k+3)/3
the possible values of k includes 3,6,9,12 .....
So this can be either multiple of 3 or any other number..
Problem solving - 1
Problem Solving - Numbers
If (3^4)(5^6)(7^3)=(35^n)(x), where x and n are both positive integers, how many different positive values of n are there?
A) 1
B) 2
C) 3
D) 4
E) 6
Answer - C
(3^4)(5^6)(7^3)=(35^n)(x)
Now 35 = 5 * 7
=> 35^n = 5^n * 7^n
3^4 * 5^3 * (5^3 * 7^3) = (35^n) * x
So x is a multiple of 3^4 * 5^3 n => 3 n is a positive integer So n=1, 2, 3
If (3^4)(5^6)(7^3)=(35^n)(x), where x and n are both positive integers, how many different positive values of n are there?
A) 1
B) 2
C) 3
D) 4
E) 6
Answer - C
(3^4)(5^6)(7^3)=(35^n)(x)
Now 35 = 5 * 7
=> 35^n = 5^n * 7^n
3^4 * 5^3 * (5^3 * 7^3) = (35^n) * x
So x is a multiple of 3^4 * 5^3 n => 3 n is a positive integer So n=1, 2, 3
Monday, June 19, 2006
Manhattan challenge problem of the week - June 19
If x is a non-zero integer, what is the value of x ^ y?
(1) x = 2
(2) (128 ^ x)[6 ^ (x + y)] = (48 ^ 2x)(3 ^ -x)
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
Answer : must be B
Statement (1) - not sufficient - value of y is still not known.
Statement (2)- Sufficient.
(128 ^ x)(6 ^ x)(6 ^ y) = [(48 ^x) (48 ^ x)] / (3 ^ x)
=> (16.8 ^ x)(6 ^ x)(6 ^ y) = (16 ^ x) (48 ^ x)
=> (16 ^ x)(8 ^ x)(6 ^ x)(6 ^ y) = (16 ^ x) (8 ^ x)(6 ^ x)
=> (6 ^ y) = 1
=> y = 0
=> (x^y) = 1 , because any number raised to the power zero is equal to 1.
Hence B must be the answer.
Official Answer and Explanation to the above problem
One of the most effective ways to begin solving problems involving exponential equations is to break down bases of the exponents into prime factors and combine exponents with the same base. Following this approach, be sure to simplify each statement as much as possible before arriving at the conclusion, since difficult problems with exponents often result in unobvious outcomes.
(1) INSUFFICIENT: While this statement gives us the value of x, we know nothing about y and cannot determine the value of x^y.
(2) SUFFICIENT: (128^x)[6^(x + y)] = (48^2x)(3^-x)
[(2^7)^x][(2 × 3)^(x + y)] = {(2^4 ) 3]^2x}(3^-x)
[(2^7)^x][2^(x + y)][3^(x + y)] = (2^8x)(3^2x)(3^-x)
[2^(8x + y)][3^(x + y)] = (2^8x)[3^(2x - x)]
(2^8x)( 2^y)(3^x)(3^y) = (2^8x)(3^x)
( 2^y)(3^y) = 1
(2 × 3)^y = 1
6^y = 1
y = 0
Since y = 0 and x is not equal to zero (as stated in the problem stem), this information is sufficient to conclude that x^y = x^0 = 1.
The correct answer is B.
(1) x = 2
(2) (128 ^ x)[6 ^ (x + y)] = (48 ^ 2x)(3 ^ -x)
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
Answer : must be B
Statement (1) - not sufficient - value of y is still not known.
Statement (2)- Sufficient.
(128 ^ x)(6 ^ x)(6 ^ y) = [(48 ^x) (48 ^ x)] / (3 ^ x)
=> (16.8 ^ x)(6 ^ x)(6 ^ y) = (16 ^ x) (48 ^ x)
=> (16 ^ x)(8 ^ x)(6 ^ x)(6 ^ y) = (16 ^ x) (8 ^ x)(6 ^ x)
=> (6 ^ y) = 1
=> y = 0
=> (x^y) = 1 , because any number raised to the power zero is equal to 1.
Hence B must be the answer.
Official Answer and Explanation to the above problem
One of the most effective ways to begin solving problems involving exponential equations is to break down bases of the exponents into prime factors and combine exponents with the same base. Following this approach, be sure to simplify each statement as much as possible before arriving at the conclusion, since difficult problems with exponents often result in unobvious outcomes.
(1) INSUFFICIENT: While this statement gives us the value of x, we know nothing about y and cannot determine the value of x^y.
(2) SUFFICIENT: (128^x)[6^(x + y)] = (48^2x)(3^-x)
[(2^7)^x][(2 × 3)^(x + y)] = {(2^4 ) 3]^2x}(3^-x)
[(2^7)^x][2^(x + y)][3^(x + y)] = (2^8x)(3^2x)(3^-x)
[2^(8x + y)][3^(x + y)] = (2^8x)[3^(2x - x)]
(2^8x)( 2^y)(3^x)(3^y) = (2^8x)(3^x)
( 2^y)(3^y) = 1
(2 × 3)^y = 1
6^y = 1
y = 0
Since y = 0 and x is not equal to zero (as stated in the problem stem), this information is sufficient to conclude that x^y = x^0 = 1.
The correct answer is B.
Saturday, June 17, 2006
DS Question - 5
What is the first term of an arithmetic progression of positive integers ?
a)Sum of the squares of the first and second term is 116.
b)The seventh term is divisible by 10.
Answer - A
Explanation - let x be the first and y be the second term.
Hence 0 less than x less than y
a)Sum of the squares of the first and second term is 116.
b)The seventh term is divisible by 10.
Answer - A
Explanation - let x be the first and y be the second term.
Hence 0 less than x less than y
From Statement 1: x^2 + y^2 = 116 only when x=2 and y=10.
Hence the first term is 4. This is the only combination that works - hence sufficient
Statement 2: let 10z be the 7th term (z is an integer)
Thus x+6(y-x)=10z => y=(10z+5x)/6. Hence the last digit of the numerator can be 5 or 0. Now y is also an integer => x must be even => least possible value of y =10.
Assuming different values of z we get
z=1; x=12 This is impossible (x should be less than y)
z=2; x=8; 8 10 12 14 16 18 20
z=3; x=6; 6 10 14 18 22 26 30 and so on -- hence insufficient
Hence the first term is 4. This is the only combination that works - hence sufficient
Statement 2: let 10z be the 7th term (z is an integer)
Thus x+6(y-x)=10z => y=(10z+5x)/6. Hence the last digit of the numerator can be 5 or 0. Now y is also an integer => x must be even => least possible value of y =10.
Assuming different values of z we get
z=1; x=12 This is impossible (x should be less than y)
z=2; x=8; 8 10 12 14 16 18 20
z=3; x=6; 6 10 14 18 22 26 30 and so on -- hence insufficient
DS Question - 4
Is the integer x divisible by 3?
1) The last digit in x is 3.
2) x+5 is divisible by 6.
Answer - B
Statement (1) is not sufficient.
If x is 33, then x is divisible by 3, and (1) holds, but if x=43, then (1) is true but x is not divisible by 3.
Statement (2) is sufficient. According to it , there exists an integer k such that x+5 = 6k => x = 6k-5 => x/3 = (6k-5)/3 = x/3 = (2k) - 5/3 => x is not divisible by 3.
Hence B is the answer i.e the (2) statement alone is sufficient to answer the question but (1) statement alone is not sufficient to answer the question.
1) The last digit in x is 3.
2) x+5 is divisible by 6.
Answer - B
Statement (1) is not sufficient.
If x is 33, then x is divisible by 3, and (1) holds, but if x=43, then (1) is true but x is not divisible by 3.
Statement (2) is sufficient. According to it , there exists an integer k such that x+5 = 6k => x = 6k-5 => x/3 = (6k-5)/3 = x/3 = (2k) - 5/3 => x is not divisible by 3.
Hence B is the answer i.e the (2) statement alone is sufficient to answer the question but (1) statement alone is not sufficient to answer the question.
Monday, June 12, 2006
Manhattan challenge problem of the week! - June 12th
If the reciprocals of two consecutive integers are added to one another, what is the sum in terms of the greater integer x?
(A) 3/x
(B) x^2 – x
(C) 2x – 1
(D) 2x – 1 /x^2 + x
(E) 2x – 1 /x^2 – x
Solution:Official answer will be known next week - i.e June 19th
Answer must be E
Greater integer = x
Consecutive smaller integer number to x = x-1
Sum of reciprocals
=1/x +1/x-1 = x-1+x/x(x-1) = 2x - 1/(x^2 - x)
Thus E is the answer.
(A) 3/x
(B) x^2 – x
(C) 2x – 1
(D) 2x – 1 /x^2 + x
(E) 2x – 1 /x^2 – x
Solution:Official answer will be known next week - i.e June 19th
Answer must be E
Greater integer = x
Consecutive smaller integer number to x = x-1
Sum of reciprocals
=1/x +1/x-1 = x-1+x/x(x-1) = 2x - 1/(x^2 - x)
Thus E is the answer.
Wednesday, June 07, 2006
Data Sufficiency -- Q No - 3
Question
Is x^2 greater than x?
1)x^2 is greater than 1.
2)x is greater than -1.
Answer- A
1) implies either x<-1 or x>1 (key is its being compared with '1' not '0' to get rid of the fractions that are <1.>
2) not sufficient. Take values 1/2 and 2 to verify.
Is x^2 greater than x?
1)x^2 is greater than 1.
2)x is greater than -1.
Answer- A
1) implies either x<-1 or x>1 (key is its being compared with '1' not '0' to get rid of the fractions that are <1.>
2) not sufficient. Take values 1/2 and 2 to verify.
Tuesday, June 06, 2006
Manhattan challenge problem of the week! - June 5th
All Squares --- Data sufficiency Question!
Is x^2 + y^2 > 4a?
(1) (x + y)^2 = 9a
(2) (x – y)^2 = a
(A) Statement (1) alone is sufficient, but statement(2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement(1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, butNEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOTsufficient.
Answer - A
Statement (1) alone is not sufficient to answer the question as - depends upon sign of x and y.
Statement (2) alone is not sufficient to answer the question as - depends upon sign of x and y.
Together, statement (1) and (2) ---
On solving (1)
Official Answer to the above problem.
(1) INSUFFICIENT: If we multiply this equation out, we get:
x2 + 2xy + y2 = 9a
If we try to solve this expression for
x2 + y2, we getx2 + y2 = 9a – 2xy
Since the value of this expression depends on the value of x and y, we don't have enough information.
(2) INSUFFICIENT: If we multiply this equation out, we get:
x2 – 2xy + y2 = a
If we try to solve this expression for x2 + y2,
we getx2 + y2 = a + 2xy
Since the value of this expression depends on the value of x and y, we don't have enough information.
(1) AND (2) INSUFFICIENT: We can combine the two expanded forms of the equations from the two statements by adding them:
x2 + 2xy + y2 = 9ax2 – 2xy + y2 = a----- 2x2 + 2y2 = 10ax2 + y2 = 5a
If we substitute this back into the original question, the question becomes: "Is 5a > 4a?"If a > 0, the answer is yes.We know from the question stem that a is nonnegative.However, if a = 0 the answer is no.
The correct answer is E.
Is x^2 + y^2 > 4a?
(1) (x + y)^2 = 9a
(2) (x – y)^2 = a
(A) Statement (1) alone is sufficient, but statement(2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement(1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, butNEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOTsufficient.
Answer - A
Statement (1) alone is not sufficient to answer the question as - depends upon sign of x and y.
Statement (2) alone is not sufficient to answer the question as - depends upon sign of x and y.
Together, statement (1) and (2) ---
On solving (1)
(x+y)^2 = 9a
x+y = + - 3 (a^1/2)
For x^2 + y^2 to be minimum x = y = + - 1.5 (a^1/2)
x^2 + y^2 = 2.25 a + 2.25 a
=4.5a > 4a
Thus answer is A
However if
a=0 then x=y=0 also, so this inequality will not hold true...
In that case it is very easy for a=x=y=0 ,no way x^2 + y^2 > 4a can be satisfied for any condition.
So answer will straight away be E.
This is assumption that x,y and a are different numbers.
Thus answer will be E in this case.
Official Answer to the above problem.
(1) INSUFFICIENT: If we multiply this equation out, we get:
x2 + 2xy + y2 = 9a
If we try to solve this expression for
x2 + y2, we getx2 + y2 = 9a – 2xy
Since the value of this expression depends on the value of x and y, we don't have enough information.
(2) INSUFFICIENT: If we multiply this equation out, we get:
x2 – 2xy + y2 = a
If we try to solve this expression for x2 + y2,
we getx2 + y2 = a + 2xy
Since the value of this expression depends on the value of x and y, we don't have enough information.
(1) AND (2) INSUFFICIENT: We can combine the two expanded forms of the equations from the two statements by adding them:
x2 + 2xy + y2 = 9ax2 – 2xy + y2 = a----- 2x2 + 2y2 = 10ax2 + y2 = 5a
If we substitute this back into the original question, the question becomes: "Is 5a > 4a?"If a > 0, the answer is yes.We know from the question stem that a is nonnegative.However, if a = 0 the answer is no.
The correct answer is E.
Tuesday, May 16, 2006
Manhattan challenge problem of the week !
A is the sum of x consecutive positive integers. b is the sum of y consecutive positive integers. For which of the following values of x and y is it impossible that a = b?
(A) x = 2; y = 6
(B) x = 3; y = 6
(C) x = 7; y = 9
(D) x = 10; y = 4
(E) x = 10; y = 7
Answer - D
Since the sum of 10 consecutive positive integers is always odd and 4 consecutive positive integers is always even, there sum a and b will never be equal, thus by above logic D is the answer.
(A) x = 2; y = 6
(B) x = 3; y = 6
(C) x = 7; y = 9
(D) x = 10; y = 4
(E) x = 10; y = 7
Answer - D
Since the sum of 10 consecutive positive integers is always odd and 4 consecutive positive integers is always even, there sum a and b will never be equal, thus by above logic D is the answer.
Friday, May 12, 2006
DS Question - 2
If d represents the hundredths digit and e represents the thousandths digit in the decimal .4de, what is the value of this decimal rounded to the nearest tenth?
(1) d – e is equal to a positive perfect square.
(2) sqrt (d) > e*e
(A) Statement (1) alone is sufficient, but statement(2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement(1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOTsufficient.
Answer - E
(1) d – e is equal to a positive perfect square.
(2) sqrt (d) > e*e
(A) Statement (1) alone is sufficient, but statement(2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement(1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOTsufficient.
Answer - E
From statement (1), we know that d – e must equal a positive perfect square. This means that d is greater than e. In addition, since any single digit minus any other single digit can yield a maximum of 9, d – e could only result in the perfect squares 9, 4, or 1.
However, this leaves numerous possibilities for the values of d and e respectively. For example, two possibilities are as follows:
d = 7, e = 3 (d – e = the perfect square 4)
d = 3, e = 2 (d – e = the perfect square 1)
In the first case, the decimal .4de would be .473, which, when rounded to the nearest tenth, is equal to .5. In the second case, the decimal would be .432, which, when rounded to the nearest tenth, is .4.
Thus, statement (1) is not sufficient on its own to answer the question.
Statement (2) tells us that sqrt d = e2. Since d is a single digit, the maximum value for d is 9, which means the maximum square root of d is 3. This means that e2 must be less than 3. Thus the digit e can only be 0 or 1.
However, this leaves numerous possibilities for the values of d and e respectively. For example, two possibilities are as follows:
d = 9, e = 1
d = 2, e = 0
In the first case, the decimal .4de would be .491, which, when rounded to the nearest tenth, is equal to .5. In the second case, the decimal would be .420, which, when rounded to the nearest tenth, is .4.
Thus, statement (2) is not sufficient on its own to answer the question.
Taking both statements together, we know that e must be 0 or 1 and that d – e is equal to 9, 4 or 1.
This leaves the following 4 possibilities:
d = 9, e = 0
d = 5, e = 1
d = 4, e = 0
d = 1, e = 0
These possibilities yield the following four decimals: .490, .451, .440, and .410 respectively. The first two of these decimals yield .5 when rounded to the nearest tenth, while the second two decimals yield .4 when rounded to the nearest tenth.
Thus, both statements taken together are not sufficient to answer the question.
The correct answer is E: Statements (1) and (2) TOGETHER are NOT sufficient.
Thursday, May 11, 2006
DS Question - 1
If x, y, and z are positive integers such that x is less than y and y is less than z, is x a factor of the odd integer z?
(1) x and y are prime numbers, whose sum is a factor of 57
(2) z is a factor of 57
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
Solution:
(1) states x+y is a factor of 57.The factors of 57 are 1,3 and 19. Since x and y are positive integers and both are prime, x+y cannot be 1, 3 because if its 1 or 3, one of x or y has to be 0 or 1 which is not a prime number.Thus x+y must be 19. Number 2 cannot be the factor of z as z is an odd number.Now x cannot be 17 as it is less than y, still even if it is, it does not tell us whether it is a factor of z or not.Thus option 1 is insufficient to answer the ques.
(2) states z is a factor of 57. Thus z is either 3 or 19. Since x is less than z, and z has factors z itself and 1, so x cannot be the factor of z, but if x = 1 it can be the factor of z, thus statement 2 alone is not sufficient to answer the question.
Now combine both 1 and 2.In this case numbers x and y are 2 and 17 as x
Nowhere it is stated that x+y=z, since z is greater than y it can be either 19 or 57, in both the cases x is not the factor of z.
Thus (1) and (2) together are sufficient to answer the question.
Hence answer is C.
Note:- You will notice that 1 and the number itself are always factors of a given number.
(1) x and y are prime numbers, whose sum is a factor of 57
(2) z is a factor of 57
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
Solution:
(1) states x+y is a factor of 57.The factors of 57 are 1,3 and 19. Since x and y are positive integers and both are prime, x+y cannot be 1, 3 because if its 1 or 3, one of x or y has to be 0 or 1 which is not a prime number.Thus x+y must be 19. Number 2 cannot be the factor of z as z is an odd number.Now x cannot be 17 as it is less than y, still even if it is, it does not tell us whether it is a factor of z or not.Thus option 1 is insufficient to answer the ques.
(2) states z is a factor of 57. Thus z is either 3 or 19. Since x is less than z, and z has factors z itself and 1, so x cannot be the factor of z, but if x = 1 it can be the factor of z, thus statement 2 alone is not sufficient to answer the question.
Now combine both 1 and 2.In this case numbers x and y are 2 and 17 as x
Nowhere it is stated that x+y=z, since z is greater than y it can be either 19 or 57, in both the cases x is not the factor of z.
Thus (1) and (2) together are sufficient to answer the question.
Hence answer is C.
Note:- You will notice that 1 and the number itself are always factors of a given number.
Mohit Gupta, one of the Gmat aspirants corrected the above explanation so correct answer is A.
But no where it has been told that X has to be odd number or cannot be even number. As X has to be a PRIME NUMBER, we can assume it to be 2. So X can be 2 and Y can be 17 and thus statement 1 if true, proves that X is not the factor of Z.Now Statement 2 in itself is not sufficient as it tells us about the possible values of Z which can be 1, 3, 19, 57 but does not tell us about the possible values of X and Y.So answer should be A.
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