BG wrote:

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

(a) take any 6 consecutive integers (+ve) for y = 10, 11, 12, 13, 14 and 15. their sum is 75. divide it by 2, we get 37.5. so take immidiate integers above and below 37.5. they are 37 and 38 and their sum is75.

(b) take any 6 consecutive integers for y = 10, 11, 12, 13, 14 and 15. their sum is 75. divide it by 3, we get 25. so put 25 in the middle. the 3 consecutive integers are 24, 25 and 26. their sum is 75.

(c) take any 9 consecutive integers for y = 3, 4, 5, 6, 7, 8, 9, 10, and 11. their sum is 63. divide it by 7, we get 9. so put 9 in the middle of 7 consecutive integers. they are 6, 7, 8, 9, 10, 11, and 12. their sum is 63.

(d) take any 10 consecutive integers for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. their sum is 55. similarly take any 4 consecutive integers and add them up, the sum is even number/integer.

here we have interesting pattren with 10 and 4 consecutive integers. any 10 such integers have sum ending at 5 where as sum of 4 consecutive integers end at even integer. so these two consecutive integers never be equal.

(e) take any 10 consecutive integers for x = 6, 7, 8, 9, 10, 11, 12, 13, 14 and 15. their sum is 105. divide it by 7, we get 15. so put 15 in the middle of 7 consecutive integers. they are 12, 13, 14, 15, 16, 17, and 18. their sum is 105.

it is D, however it is lengthy.