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x is the sum of y consecutive integers

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by GMATsid2016 » Thu Dec 01, 2016 8:49 am
x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

A) x = w

B) x > w

C) x/y is an integer

D) w/z is an integer

E) x/z is an integer

OAC
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by DavidG@VeritasPrep » Thu Dec 01, 2016 8:59 am
GMATsid2016 wrote:x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

A) x = w

B) x > w

C) x/y is an integer

D) w/z is an integer

E) x/z is an integer

OAC
Pick easy numbers. Say z = 1. y = 2z, so y = 2.

x is the sum of 2 consecutive integers, so say x = 1 + 2 = 3.

w is the sum of 1 consecutive integer, so it can be any number we want.

To summarize: z = 1, y = 2, x = 3, and w = anything we want.

A) That can be true. Just say w = 2
B) That can be true. Just say w = 3
C) 3/2 : no good
D) That can be true: if z is 1, w/z will always be an integer
E) That can be true: if z is 1, x/z will always be an integer

If A, B, D, and E could all be true, the answer must be C
Last edited by DavidG@VeritasPrep on Thu Dec 01, 2016 9:00 am, edited 1 time in total.
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by GMATGuruNY » Thu Dec 01, 2016 9:00 am
x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

A)x = w
B)x > w
C)x/y is an integer
D)w/z is an integer
E)x/z is an integer
w, x, y and z are all integers.
Test the SMALLEST POSSIBLE CASE.
Let z=1.

Since z=1, it must be possible that w/z and x/z are integers.
Eliminate D and E.

Since z=1, w is the sum of 1 consecutive integer, implying that w can be ANY INTEGER.
Thus, it must be possible that x=w or that x>w.
Eliminate A and B.

The correct answer is C.
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by DavidG@VeritasPrep » Thu Dec 01, 2016 9:31 am
GMATsid2016 wrote:x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

A) x = w

B) x > w

C) x/y is an integer

D) w/z is an integer

E) x/z is an integer

OAC
You could also use some fun number properties.

Rule 1: The sum of y consecutive integers will always be a multiple of y if y is ODD.
Rule 2: The sum of y consecutive integers will never be a multiple of y if y is EVEN.

If y = 2z, where z is an integer, we know y is EVEN. If x is the sum of y consecutive integers, and y is EVEN, then according to rule 2, x cannot be a multiple of y. If integer x is not a multiple of integer y, x/y can't be an integer. Answer is C
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