Question 1

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oquiella: Master | Next Rank: 500 Posts; Posts: 164; Joined: Tue May 12, 2015 9:06 am; Thanked: 3 times

by oquiella » Fri Sep 18, 2015 4:47 am

1. If x,y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is not necessarily an integer

A. x+z/z

B. y+z/x

C. x+y/z

D. xy/z

E. yz/x

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Source: Beat The GMAT — Problem Solving | Fri Sep 18, 2015 4:47 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Fri Sep 18, 2015 4:59 am

Our goal is to find the answer choice that does NOT have to be an integer.
Since z is a factor of x and x is a factor of y, z divides into both x and y.
Since z is divides into both x and y, the answer choices with z in the denominator (A, C and D) are likely to yield integer values.

Try to show that B does not have to be an integer.
Let z=1, x=2, y=4.
B: (y+z)/x = (4+1)/2 = 5/2. Not an integer.

The correct answer is B.

Last edited by GMATGuruNY on Fri Sep 18, 2015 1:11 pm, edited 1 time in total.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Fri Sep 18, 2015 7:38 am

Note: To avoid confusion, I added some brackets to your answer choice.

If x, y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT necessarily an integer?

A) (x+z)/z
B) (y+z)/x
C) (x+y)/z
D) (xy)/z
E) (yz)/x

Mitch already showed the approach of looking for values of x, y and z, such that we get a non-integer.

Here's another approach:

x is a multiple of z
So, we can say that x = kz (for some integer k)

x is a factor of y
In other words, y is a multiple of x.
So, we can say that y = jx (for some integer j)
IMPORTANT: Since we already know that x = kz, let's replace x with kz to get: y = jkz

So, x = kz, y = jkz and z = z.
Let's plug in these values to see what we get:

A) (x+z)/z = (kz+z)/z = k+1 (= INTEGER)
B) (y+z)/x = (jkz+z)/kz = (jk+1)/k (not necessarily an integer)
C) (x+y)/z = (kz+jkz)/z = k+jk (= INTEGER)
D) (xy)/z = [(kz)(jkz)]/z = jkÂ²z (= INTEGER)
E) (yz)/x = [(jkz)(z)]/kz = jz (= INTEGER)

Answer = B

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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by [email protected] » Fri Sep 18, 2015 9:39 am

Hi oquiella,

Both Mitch and Brent have provided solutions to this question, so I won't rehash any of that math here. Instead, I want to point out a design 'shortcut' in this question:

Notice how the prompt asks 'which of the following is NOT necessarily an integer?" - this wording implies that 4 of the answers will ALWAYS be integers, while one will not always be an integer. Once we find the one answer that isn't necessarily an integer, then we can stop working - that answer WILL be the correct answer.

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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