The remainder, when a number n

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aaron1981: Junior | Next Rank: 30 Posts; Posts: 10; Joined: Tue Oct 15, 2013 4:45 am

by aaron1981 » Tue Jun 13, 2017 11:32 pm

The remainder, when a number n is divided by 6, is p. The remainder, when the same number n is divided by 12, is q. Is p < q?

(1) n is a positive number having 8 as a factor.
(2) n is a positive number having 6 as a factor.

OA C

Pl. help me with algebraic way as well as plug-in value way.

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Source: Beat The GMAT — Data Sufficiency | Tue Jun 13, 2017 11:32 pm

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by Jay@ManhattanReview » Wed Jun 14, 2017 1:57 am

aaron1981 wrote:The remainder, when a number n is divided by 6, is p. The remainder, when the same number n is divided by 12, is q. Is p < q?

(1) n is a positive number having 8 as a factor.
(2) n is a positive number having 6 as a factor.

OA C

Pl. help me with algebraic way as well as plug-in value way.

Say n = 6k + p; where k is any integer and p is any value among 1, 2, 3, 4, and 5.
Also, say n = 12m + q; where m is any integer and q is any value among 1, 2, 3, 4, ..., and 11.

We have to deduce whether p < q.

Statement 1: n is a positive number having 8 as a factor.

Since n is a positive number having 8 as a factor, n is one among 8, 18, 24, 32, 40, ...

Case 1: If n = 8, the remainder when n = 8 divided by 6 is p = 2, and the remainder when n = 8 divided by 12 is q = 8.

p = 2 < q = 8. The answer is YES.

Case 2: If n = 24, the remainder when n = 24 divided by 6 is p = 0, and the remainder when n = 24 divided by 12 is q = 0.

p = q = 0. The answer is No. No unique answer.

Statement 2: n is a positive number having 6 as a factor.

Since n is a positive number having 6 as a factor, n is one among 6, 12, 18, 24, ...

Case 1: If n = 6, the remainder when n = 6 divided by 6 is p = 0, and the remainder when n = 6 divided by 12 is q = 6.

p = 0 < q = 6. The answer is YES.

Case 2: If n = 24, the remainder when n = 24 divided by 6 is p = 0, and the remainder when n = 24 divided by 12 is q = 0.

p = q = 0. The answer is No. No unique answer.

Statement 1 & 2: n is a positive number having 6 as well as 8 as a factor.

=> n is a positive number having LCM of 6 and 8 as a factor.

LCM of 6 & 8 = 24.

=> n = 24k; k is any integer

Since 24 is divisible by 6 and 8, the reminders would be 0, thus p = q = 0. The answer is NO. A unique answer.

The correct answer: C

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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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 » Wed Jun 14, 2017 2:03 am

aaron1981 wrote:The remainder, when a number n is divided by 6, is p. The remainder, when the same number n is divided by 12, is q. Is p < q?

(1) n is a positive number having 8 as a factor.
(2) n is a positive number having 6 as a factor.

Statement 1:
Case 1: n=8
Since n/6 = 8/6 = 1 R2, p=2.
Since n/12 = 8/12 = 0 R8, q=8.
In this case, p<q, with the result that the answer to the question stem is YES.

Case 2: n=16
Since n/6 = 16/6 = 2 R4, p=4.
Since n/12 = 16/12 = 1 R4, q=4.
In this case, p=q, with the result that the answer to the question stem is NO.
INSUFFICIENT.

Statement 2:
Case 3: n=6
Since n/6 = 6/6 = 1 R0, p=0.
Since n/12 = 6/12 = 0 R6, q=6.
In this case, p<q, with the result that the answer to the question stem is YES.

Case 3: n=12
Since n/6 = 12/6 = 2 R0, p=0.
Since n/12 = 12/12 = 1 R0, q=0.
In this case, p=q, with the result that the answer to the question stem is NO.
INSUFFICIENT.

Statements combined:
Since n must be divisible by 8 and 6, n must be a multiple of 24:
24, 48, 72...
Dividing a multiple of 12 by 6 will yield a remainder of 0.
Thus, p=0.
Dividing a multiple of 24 by 12 will also yield a remainder of 0.
Thus, q=0.
Since p=q, the answer to the question stem is NO.
SUFFICIENT.

The correct answer is C.

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