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The positive integer n is greater than 10.

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by NandishSS » Tue Apr 11, 2017 7:27 am
The positive integer n is greater than 10. What is the remainder when the positive integer n is divided by 12?

(1) n is 20 more than a multiple of 36.

(2) n is 4 less than a multiple of 3 and n is 4 less than a multiple of 8.

OA:D
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by DavidG@VeritasPrep » Tue Apr 11, 2017 8:04 am
NandishSS wrote:The positive integer n is greater than 10. What is the remainder when the positive integer n is divided by 12?

(1) n is 20 more than a multiple of 36.

(2) n is 4 less than a multiple of 3 and n is 4 less than a multiple of 8.

OA:D
Statement 1: if we list out multiples of 36, we can then simply add 20 to each to generate a list of testable values: 36, 72, 108. Add 20 to get 56, 92, 128.
56/12 --> remainder = 8
92/12 --> remainder = 8
128/12 --> remainder = 8

No matter what the remainder is 8. This statement alone is sufficient.

(Note that one could rephrase statement 1 as: when n is divided by 36, there is a remainder of 20. Because 36 is a multiple of 12, if we know the remainder when n is divided by 36, we'd be able to determine the remainder when n is divided by 12.)

Statement 2: The LCM of 3 and 8 is 24. So we can rephrase the statement to stipulate that n is 4 less than a multiple of 24. Multiples of 24 = 24, 48, 72, 96. Subtract 4 to get 20, 44, 68, 92

20/12 --> remainder = 8
44/12 --> remainder = 8
68/12 --> remainder = 8

(Note again that 24 is a multiple of 12, a clue that this statement will be sufficient on its own.)

This statement alone is sufficient. The answer is D
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by Brent@GMATPrepNow » Tue Apr 11, 2017 10:50 am
NandishSS wrote:The positive integer n is greater than 10. What is the remainder when the positive integer n is divided by 12?

(1) n is 20 more than a multiple of 36.
(2) n is 4 less than a multiple of 3 and n is 4 less than a multiple of 8.
Target question: What is the remainder when the positive integer n is divided by 12?

Statement 1: n is 20 more than a multiple of 36.
In other words, n = 36k + 20 for some integer k
Let's do some rewriting.
n = 36k + 20
= (12)(3)k + 12 + 8
= 12(3k + 1) + 8
= 12(some integer) + 8
Here, we can see that n equals 8 more than some multiple of 12
This means that, if we divide n by 12, the remainder will be 8
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: n is 4 less than a multiple of 3 and n is 4 less than a multiple of 8
Let's examine both pieces separately:

n is 4 less than a multiple of 3
In other words, n = 3k - 4 for some integer k
If we add 4 to both sides, we get n + 4 = 3k
This means that n+4 is a multiple of 3

n is 4 less than a multiple of 8
In other words, n = 8j - 4 for some integer j
If we add 4 to both sides, we get n + 4 = 8j
This means that n+4 is a multiple of 8

If n+4 is a multiple of 3 AND 8, then we can conclude that n+4 is a multiple of 24
ALSO, if n+4 is a multiple of 24, then we can conclude that n+4 is a multiple of 12
Finally, if if n+4 is a multiple of 12, we can write: n+4 = 12q for some integer q

We can also write: n+4 = 12q - 12 + 12 [you'll see why shortly]
n+4 = 12(q - 1) + 12
n = 12(q - 1) + 8
n = 12(some integer) + 8
Once again, we can see that n equals 8 more than some multiple of 12
This means that, if we divide n by 12, the remainder will be 8
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent
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by Jay@ManhattanReview » Tue Apr 11, 2017 8:38 pm
NandishSS wrote:The positive integer n is greater than 10. What is the remainder when the positive integer n is divided by 12?

(1) n is 20 more than a multiple of 36.

(2) n is 4 less than a multiple of 3 and n is 4 less than a multiple of 8.

OA:D
Hi NandishSS,

We have n > 10.

Question: What is the remainder when the positive integer n is divided by 12?

S1: n is 20 more than a multiple of 36.

=> n = 36k + 20; where k is a non-negative integer.

Since 36 is completely divisible by 12, the remainder would be decided by 20 divided by 12. Thus, the remainder when 20 is divided by 12 = 8. Sufficient.

S2: n is 4 less than a multiple of 3 and n is 4 less than a multiple of 8.

=> n = 3k - 4; where k is a positive integer greater than 4; we have to ensure that n > 10.

=> n = 8m - 4; where m is a positive integer greater than 1; we have to ensure that n > 10.

=> 3k - 4 = 8m - 4

=> 3k = 8m

=> The minimum value of k = 8 and the minimum value of m = 3.

=> n = 3k - 4 = 3*8 - 4 = 20

Thus, the remainder when 20 is divided by 12 = 8. Sufficient.

The correct answer: D

Hope this helps!

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