If x = 3 ^ 21 and y = 6 ^ 55, what is the remainder when xy is divided by 10?
(A) 2
(B) 3
(C) 4
(D) 6
(E) 8
Answer : The correct answer is E.
Since every multiple of 10 must end in zero, the remainder from dividing xy by 10 will be equal to the units’ digit of xy. In other words, the units’ digit will reflect by how much this number is greater than the nearest multiple of 10 and, thus, will be equal to the remainder from dividing by 10. Therefore, we can rephrase the question: “What is the units’ digit of xy?”
Next, let’s look for a pattern in the units’ digit of 3 ^ 21. Remember that the GMAT will not expect you to do sophisticated computations; therefore, if the exponent seems too large to compute, look for a shortcut by recognizing a pattern in the units' digits of the exponent:
3 ^ 1 = 3
3 ^ 2 = 9
3 ^ 3 = 27
3 ^ 4 = 81
3 ^ 5 = 243
As you can see, the pattern repeats every 4 terms, yielding the units digits of 3, 9, 7, and 1. Therefore, the exponents 3 ^ 1, 3 ^ 5, 3 ^ 9, 3 ^ 13, 3 ^ 17, and 3 ^ 21 will end in 3, and the units’ digit of 3 ^ 21 is 3.
Next, let’s determine the units’ digit of 6 ^ 55 by recognizing the pattern:
6 ^ 1 = 6
6 ^ 2 = 36
6 ^ 3 = 256
6 ^ 4 = 1,296
As shown above, all positive integer exponents of 6 have a units’ digit of 6. Therefore, the units' digit of 6 ^ 55 will also be 6.
Finally, since the units’ digit of 3 ^ 21 is 3 and the units’ digit of 6 ^ 55 is 6, the units' digit of 3 ^ 21 × 6 ^ 55 will be equal to 8, since 3 × 6 = 18.
Therefore, when this product is divided by 10, the remainder will be 8.
Thursday, January 25, 2007
Manhattan Challenge Problem of the week ! - Jan 15
Sn = 2Sn-1 + 4 and Qn = 4Qn-1 + 8 for all n > 1. If S5 = Q4 and S7 = 316, what is the first value of n for which Qn is an integer?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Answer : C is the correct answer.
If S7 = 316, then 316 = 2S6 + 4, which means that S6=156.
We can then solve for S5:
156 = 2S5 + 4, so S5 = 76
Since S5 = Q4, we know that Q4 = 76 and we can now solve for previous Qn’s to find the first n value for which Qn is an integer.
To find Q3: 76 = 4Q3 + 8, so Q3 = 17
To find Q2: 17 = 4Q2 + 8, so Q2 = 9/2
It is clear that Q1 will also not be an integer so there is no need to continue.
Q3 (n = 3) is the first integer value.
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Answer : C is the correct answer.
If S7 = 316, then 316 = 2S6 + 4, which means that S6=156.
We can then solve for S5:
156 = 2S5 + 4, so S5 = 76
Since S5 = Q4, we know that Q4 = 76 and we can now solve for previous Qn’s to find the first n value for which Qn is an integer.
To find Q3: 76 = 4Q3 + 8, so Q3 = 17
To find Q2: 17 = 4Q2 + 8, so Q2 = 9/2
It is clear that Q1 will also not be an integer so there is no need to continue.
Q3 (n = 3) is the first integer value.
Tuesday, January 23, 2007
Manhattan Challenge Problem of the week ! - Jan 1
"Smurfs, Elves and Fairies"
One smurf and one elf can build a treehouse together in two hours, but the smurf would need the help of two fairies in order to complete the same job in the same amount of time. If one elf and one fairy worked together, it would take them four hours to build the treehouse. Assuming that work rates for smurfs, elves, and fairies remain constant, how many hours would it take one smurf, one elf, and one fairy, working together, to build the treehouse?
(A) 5/7
(B) 1
(C) 10/7
(D) 12/7
(E) 22/7
Answer : D is the correct answer.
The combined rate of individuals working together is equal to the sum of all the individual working rates.
Let s = rate of a smurf, e = rate of an elf, and f = rate of a fairy. A rate is expressed in terms of treehouses/hour. So for instance, the first equation below says that a smurf and an elf working together can build 1 treehouse per 2 hours, for a rate of 1/2 treehouse per hour.
1) s + e = 1/2
2) s + 2 f = 1/2
3) e + f = 1/4
The three equations can be combined by solving the first one for s in terms of e, and the third equation for f in terms of e, and then by substituting both new equations into the middle equation.
1) s = 1/2 – e
2) (1/2 – e) + 2 (1/4 – e) = 1/2
3) f = 1/4 – e
Now, we simply solve equation 2 for e:
(1/2 – e) + 2 (1/4 – e) = 1/2
2/4 – e + 2/4 – 2 e = 2/4
4/4 – 3e = 2/4
-3e = -2/4
e = 2/12
e = 1/6
Once we know e, we can solve for s and f:
s = 1/2 – e
s = 1/2 – 1/6
s = 3/6 – 1/6
s = 2/6s = 1/3
f = 1/4 – e
f = 1/4 – 1/6
f = 3/12 – 2/12
f = 1/12
We add up their individual rates to get a combined rate:
e + s + f
=1/6 + 1/3 + 1/12
=2/12 + 4/12 + 1/12
= 7/12
Remembering that a rate is expressed in terms of treehouses/hour, this indicates that a smurf, an elf, and a fairy, working together, can produce 7 treehouses per 12 hours. Since we want to know the number of hours per treehouse, we must take the reciprocal of the rate. Therefore we conclude that it takes them 12 hours per 7 treehouses, which is equivalent to 12/7 of an hour per treehouse.
The correct answer is D.
One smurf and one elf can build a treehouse together in two hours, but the smurf would need the help of two fairies in order to complete the same job in the same amount of time. If one elf and one fairy worked together, it would take them four hours to build the treehouse. Assuming that work rates for smurfs, elves, and fairies remain constant, how many hours would it take one smurf, one elf, and one fairy, working together, to build the treehouse?
(A) 5/7
(B) 1
(C) 10/7
(D) 12/7
(E) 22/7
Answer : D is the correct answer.
The combined rate of individuals working together is equal to the sum of all the individual working rates.
Let s = rate of a smurf, e = rate of an elf, and f = rate of a fairy. A rate is expressed in terms of treehouses/hour. So for instance, the first equation below says that a smurf and an elf working together can build 1 treehouse per 2 hours, for a rate of 1/2 treehouse per hour.
1) s + e = 1/2
2) s + 2 f = 1/2
3) e + f = 1/4
The three equations can be combined by solving the first one for s in terms of e, and the third equation for f in terms of e, and then by substituting both new equations into the middle equation.
1) s = 1/2 – e
2) (1/2 – e) + 2 (1/4 – e) = 1/2
3) f = 1/4 – e
Now, we simply solve equation 2 for e:
(1/2 – e) + 2 (1/4 – e) = 1/2
2/4 – e + 2/4 – 2 e = 2/4
4/4 – 3e = 2/4
-3e = -2/4
e = 2/12
e = 1/6
Once we know e, we can solve for s and f:
s = 1/2 – e
s = 1/2 – 1/6
s = 3/6 – 1/6
s = 2/6s = 1/3
f = 1/4 – e
f = 1/4 – 1/6
f = 3/12 – 2/12
f = 1/12
We add up their individual rates to get a combined rate:
e + s + f
=1/6 + 1/3 + 1/12
=2/12 + 4/12 + 1/12
= 7/12
Remembering that a rate is expressed in terms of treehouses/hour, this indicates that a smurf, an elf, and a fairy, working together, can produce 7 treehouses per 12 hours. Since we want to know the number of hours per treehouse, we must take the reciprocal of the rate. Therefore we conclude that it takes them 12 hours per 7 treehouses, which is equivalent to 12/7 of an hour per treehouse.
The correct answer is D.
Manhattan Challenge Problem of the Week ! - dec 18
Chicago Rain
If the probability of rain on any given day in Chicago during the summer is 50%, independent of what happens on any other day, what is the probability of having exactly 3 rainy days from July 4 through July 8, inclusive?
(A) 1/32
(B) 2/25
(C) 5/16
(D) 8/25
(E) 3/4
Answer: Correct ans is C. For OE click on the link below.
Manhattan Challenge Problem
If the probability of rain on any given day in Chicago during the summer is 50%, independent of what happens on any other day, what is the probability of having exactly 3 rainy days from July 4 through July 8, inclusive?
(A) 1/32
(B) 2/25
(C) 5/16
(D) 8/25
(E) 3/4
Answer: Correct ans is C. For OE click on the link below.
Manhattan Challenge Problem
Manhattan Challenge Problem of the week ! - dec 25
Is n less than 1 ?
(1) nx – n less than 0
(2) x–1 = –2
Answer : C
From stat(1): (n^x) – n less than 0 -- no information about x --- insufficient
From stat(2): x^–1 = –2 -- no information about n --- insufficient
Taking both the statements together we get: From stat 2 we get x = -1/ 2
Substituting value of x in stat 1 we get (n^-1/2) -n less than 0
(n^3/2)>1
=> n will be greater than 1 ---- sufficient
Hence C
(1) nx – n less than 0
(2) x–1 = –2
Answer : C
From stat(1): (n^x) – n less than 0 -- no information about x --- insufficient
From stat(2): x^–1 = –2 -- no information about n --- insufficient
Taking both the statements together we get: From stat 2 we get x = -1/ 2
Substituting value of x in stat 1 we get (n^-1/2) -n less than 0
(n^3/2)>1
=> n will be greater than 1 ---- sufficient
Hence C
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