12 Days of Christmas GMAT Competition - Day 9: If m and n are positive

Question

12 Days of Christmas GMAT Competition with Lots of Fun

If m and n are positive integers, do they yield the same remainder when divided by 24?

(1) m and n yield the same remainder when divided by 8.
(2) |m - n| is a multiple of 9

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M36-93

Deepakjhamb · Answer

If m and n are positive integers, do they yield the same remainder when divided by 24?

(1) m and n yield the same remainder when divided by 8.
(2) |m - n| is a multiple of 9

Solution : 
answer is c)

lets analyze stem first

lets assume both m and n leave same remainder when divided by 24

so we can write m= 24 X + r and n as 24y + r where r >=0 and < 24

now |m -n | = 24 |X- Y| so basically question is asking if |m-n| = multiple of 24

statement 1 : gives us that |m-n|= 8k

so obviously we can have yes when when k is multiple of 3 and no if its not a multiple of 3 for the q , so insufficient

statement 2: we are given that |m-n|= 9K

so obviously we can have yes when k = multiple of 8 and no if k is not a multiple of 8 so insufficient

when we combine statement 1 and statement 2 we get |m-n| = 72 K

So that's perefect if |m-n| is multiple of 72 , it will definitely be a multiple of 24

so answer goes with c) Both statements combined together are sufficient but neither statement alone is sufficient

TestPrepUnlimited · Answer

Statement 1: 
The difference of m and n must be a multiple of 8, so we can write  |m - n| = 8*i . Thus they may have a difference of 8, 16, 24 etc and they can have different remainders when divided by 24. Insufficient.

Statement 2: 
Similar as above, if  i =24  then they have the same remainder. Otherwise different remainders. Insufficient.

Combined: 
Combined we know their difference is a multiple of 8 and a multiple of 9. Then their difference must be a multiple of LCM(8, 9) = 72, and thus when we divide by 24 (which is factor of 72) they must have the same remainder. Sufficient.

Ans: C

zhanbo · Answer

My answer is  (C) : We need  both  statement in order to answer the question, and the answer is they do yield the same remainder.

(1) m and n yield the same remainder when divided by 8. 
If m = n = 1, they yield the same remainder when divided by 8, and they also yield the same remainder when divided by 24.
If m = 1, n = 9, they yield the same remainder when divided by 8, but they do not yield the same remainder when divided by 24.
NOT SUFFICIENT

(2) |m - n| is a multiple of 9 
If m =1, n = 10, |m - n| is a multiple of 9, but they do not yield the same remainder when divided by 24. (Because 0 is a multiple of 9, I can actually reconsider m = n = 1 here.)
If m =1, n = 1 + 9 * 24 = 217, |m - n| is a multiple of 9, and they also yield the same remainder when divided by 24.
NOT SUFFICIENT

By taking (1）and (2) together , we know m and n should have a relationship that looks like：
m = n + 72 * 1 (*2, *3, ...) 
or n - 72 * 1 (*2, *3, ...) 
Because 72 is dividable by 24, we can solve the question: they do yield the same remainder when divided by 24.
SUFFICIENT

Bunuel · Answer

Official Solution:

If  m  and  n  are positive integers, do they yield the same remainder when divided by 24? 
 
If the remainders were the same then we'd be able to write  m=24p+r  and  n=24q+r .
 
Subtract one from another  m-n=24(p-q) . So, when  m  and  n  yield the same remainder when divided by 24, their difference is a multiple of 24.
 
(1)  m  and  n  yield the same remainder when divided by 8. 
 
The same way as above, we can deduce that this statement implies that  m-n  is a multiple of 8. So,  m-n  may or may not be a multiple of 24. Not sufficient.
 
(2)  |m - n|  is a multiple of 9.
 
This means that  m-n  is a multiple of 3 but we don't know whether it is also a multiple of 8. Not sufficient.
 
(1)+(2) From (1)  m-n  is a multiple of 8 and from (2) it's is a multiple of 3, thus  m-n  is a multiple of 24. Sufficient.

Answer: C

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12 Days of Christmas GMAT Competition - Day 9: If m and n are positive

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