If a is the sum of x consecutive positive integers. b is the sum of y

Question

If 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

i'm looking for a better way of solving the question

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OA ... D

If 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?

Lets assume first term be A and B for respective series.
a = 10A + 45 = Odd Number
b = 4B + 6 = Even Number

a = b Cannot be

10A + 39 = 4B
We cannot get a number, which has unit digit 9 and multiple of 4
Hence a=b impossible.

Bunuel · Answer

The sum n consecutive integers is give by: Sum=\frac{(2a_1+n-1)*n}{2} (check Number Theory chapter of Math Book for more: math-number-theory-88376.html); Note that: when n=even=2*odd , so when n (# of consecutive integers) is even but not a multiple 4 then Sum=\frac{(2a_1+n-1)*n}{2}=\frac{(even+even-odd)*(2*odd)}{2}=odd*odd=odd ; when n=even=2*even , so when n is a multiple 4 then Sum=\frac{(2a_1+n-1)*n}{2}=\frac{(even+even-odd)*(2*even)}{2}=odd*even=even ; That's because a set of even number of consecutive integers has half even and half odd terms. The sum of even terms is obviously even. As for odd terms: their sum is even if their number is even (so total # of terms is multiple of 4) and their sum is odd if their number is odd (so total number of terms is even but not a multiple of 4); So, the sum of 10 (not a multiple of 4) consecutive integers will be odd (the sum of 5 even and 5 odd integers) and the sum of 4 (multiple of 4) consecutive integers will be even (the sum of 2 even and 2 odd integers), so option D is not possible. Answer: D. Hope it's clear.

dimitri92 · Answer

v.nice explanation ... +1 kudos

gmatkillah · Answer

I approached this in a different way, although may be longer but it's certainly more intuitive

It is impossible that a = b if one of them is necessarily even while the other is necessarily odd

for two consecutive integers the sum is necessarily odd
for six consecutive integers the sum is necessarily odd (sum of 3 "two consecutive integers" which are all odd)
Thus A is incorrect.

The sum of 3 consecutive integers can be even or odd (an odd integer from the first 2 + an additional integer which can be even or odd). Thus B is incorrect. You can do a similar approach for C and E (it follows that the sum of a specified odd number of consecutive integers can be odd or even).

Leaving us with D to be the only not-wrong answer. To confirm, you can easily deduce that the sum of 4 consecutive integers (sum of 2 odd integers) is necessarily even while the sum of 10 consecutive integers (sum of 5 odd integers) is necessarily odd. Thus D is the right answer.

Bunuel · Answer

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-a-is-the-sum-of-x-consecutive-positive-integers-b-is-the-128656.html

If a is the sum of x consecutive positive integers. b is the sum of y

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