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

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if m>n, then is mn divisible by 3?  [#permalink]

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26 Sep 2017, 05:51
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65% (hard)

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46% (02:33) correct 54% (01:56) wrong based on 35 sessions

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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.

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if m>n, then is mn divisible by 3?  [#permalink]

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Updated on: 26 Sep 2017, 07:17
fitzpratik wrote:
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.

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 $$mn=[3(k+q)+4]*[3(k-q)+1] = 9(k^2-q^2)+12(k-q)+3(k+q)+4$$

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

Option C

Originally posted by niks18 on 26 Sep 2017, 07:13.
Last edited by niks18 on 26 Sep 2017, 07:17, edited 1 time in total.
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Re: if m>n, then is mn divisible by 3?  [#permalink]

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26 Sep 2017, 07:16
niks18 wrote:
fitzpratik wrote:
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.

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={3(k+q)+4}*{3(k-q)+1} = 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

Got it! Thanx.
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if m>n, then is mn divisible by 3?  [#permalink]

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29 Sep 2017, 04:14
fitzpratik wrote:
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.

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....
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Re: if m>n, then is mn divisible by 3?  [#permalink]

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19 Dec 2018, 05:16
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).
Re: if m>n, then is mn divisible by 3?   [#permalink] 19 Dec 2018, 05:16
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# if m>n, then is mn divisible by 3?

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