if mn, then is mn divisible by 3?

Question

If  m>n , then is  mn  divisible by 3?

1. The remainder when m+n is divided by 6 is 5.

2. The remainder when m-n is divided by 6 is 3.

niks18 · Answer

We need the value of both  m  &  n  to solve the question

Statement 1 : let  m+n=6k+5 -----------------------(1)

from this equation we cannot find the value of  m  or  n . Hence  Insufficient

Statement 2 : let  m-n=6q+3 ------------------------(2)

from this equation we cannot find the value of  m  or  n . Hence  Insufficient

Combining 1 & 2: add equation 1 & 2

so  2m=6k+6q+8  or  m=3(k+q)+4

substitute the value of  m  in equation 1 to get the value of  n=3(k-q)+1

so  mn= *  = 9(k^2-q^2)+12(k-q)+3(k+q)+4

Hence on dividing  mn  by  3  remainder  1  will be left.  Sufficient

Option  C

Apppyyyiiieee · Answer

Got it! Thanx.

CriticalCritic · Answer

Alternative solution...

The question implicitly asks whether M or N or both are multiples of 3. So I approach it that way.

Also, observe that either of the given remainders (doesn't matter which) are odd after being divided by 6. Therefore, one of M or N must be odd, and the other must be even.

All even multiples of 3 are also multiples of 6. And all odd are multiples of 3 (duh). So let's say M may be the multiple of 6 (6m), and N may be the *ODD* multiple of 3 (3n, again note that n must be an odd number).

First consider whether M=6m. We see here that the second answer option gives a remainder of 3. The only possible way for M to be a multiple of 6 that, after being divided by 6, results in a remainder of 3, is if N=3n. However, under no circumstances will this ever result in a remainder of 5. Therefore, under no circumstances can M=6m.

So, can we be certain whether N=3n? Fist recall that N must be an even number and is NOT a multiple of 3. Here it IS possible that a remainder of 5 can be achieved. However, using the same prior logic, it's not possible to obtain a remainder of 3, after M is divided by 6, unless M=6m.

Therefore, using both A and B, we have sufficient info to show that neither M or N can be a multiple of 3. Sufficient.

...now rereading, I think my explanation is about as clear as mud....

atrayee345 · Answer

m+n gives 5 as a remainder when divided by 6. So, the minimum value of m+n is 5.
Similarly, minimum value of m-n=3.
So, m+n= 5
m-n= 3
Solving both the equations we get, m=4, n=1
Hence, both the statements are required to solve the problem. Answer= (C).

if m>n, then is mn divisible by 3?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions