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.

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

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.

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 is less than y and both x and y are prime numbers, now according to option 2, z is either 1,3, 19 or 57.

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.