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If p and q are integers

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Source: — Data Sufficiency |
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by GMATGuruNY » Tue Jun 20, 2017 5:07 am
aaron1981 wrote:If p and q are integers, can (q-1) always be expressed as an integer multiple of p?

(1) p > q
(2) q > 1
Does (q-1)/p = integer?

Statement 1: p>q
If p=2 and q=1, then (q-1)/p = 0/2 = 0, so the answer to the question stem is YES.
If p=3 and q=2, then (q-1)/p = (2-1)/3 = 1/3, so the answer to the question stem is NO.
INSUFFICIENT.

Statement 2: q>1
No information about p.
INSUFFICIENT.

Statements combined: p>q>1.
Since q>1, q-1 > 0, implying that q-1 is a POSITIVE INTEGER.
Since p>q, p is a POSITIVE INTEGER greater than q and thus greater than q-1.
Implication:
(q-1)/p = (smaller positive integer)/(greater positive integer) = fraction.
Thus, the answer to the question stem is NO.
SUFFICIENT.

The correct answer is C.
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by Jay@ManhattanReview » Tue Jun 20, 2017 5:13 am
aaron1981 wrote:If p and q are integers, can (q-1) always be expressed as an integer multiple of p?

(1) p > q
(2) q > 1

OA C
We need to determine whether:

q - 1 ƒ= kp, where k is an integer

Statement 1:

We know that p > q:

Case 1: Let p ƒ= 2, q ƒ= - 5 ƒ=> q - 1 ƒ= -6
ƒ=> q - 1 ƒ= (-3…) x p. The answer is 'Yes.'

Case 2: Let p ƒ= 4, q ƒ= 3 ƒ=> q - 1 ƒ= 2
ƒ=> q - 1 ƒ= (1/2…) x p. The answer is 'No.'

Thus, there is no unique answer. - Insufficient

Statement 2:

There is no information about p. - Insufficient

Statement 1 & 2 together:

We have: p > q > 1

Thus, p and q are positive integers greater than 1.

Thus, (q - 1) must be less than p, and both (q - 1) and p are positive.

q - 1 = ƒ kp ƒ=> (q - 1)/p = k < 1

=> (q - 1) cannot be expressed as an integer multiple of p. - Sufficient

The correct answer: C

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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