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If m and n are positive integers...

tagged by: swerve

This topic has 1 expert reply and 0 member replies

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swerve Master | Next Rank: 500 Posts
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If m and n are positive integers...

Wed Nov 08, 2017 7:10 am
If m and n are positive integers such that m>n, what is the remainder when m^2-n^2 is divided by 21?

1) The remainder when (m+n) is divided by 21 is 1.
2) The remainder when (m-n) is divided by 21 is 1.

The OA is C.

Please, can any expert explain this DS question for me? I have many difficulties to understand why that is the correct answer. Thanks.

GMAT/MBA Expert

Jay@ManhattanReview GMAT Instructor
Joined
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Wed Nov 08, 2017 11:09 pm
swerve wrote:
If m and n are positive integers such that m>n, what is the remainder when m^2-n^2 is divided by 21?

1) The remainder when (m+n) is divided by 21 is 1.
2) The remainder when (m-n) is divided by 21 is 1.

The OA is C.

Please, can any expert explain this DS question for me? I have many difficulties to understand why that is the correct answer. Thanks.
We have to get the remainder when (m^2 - n^2) is divided by 21.

1) The remainder when (m+n) is divided by 21 is 1.

Say m + n = 21a + 1; where a is any integer (quotient)

We cannot conclude on this basis whether (m^2 - n^2) is divisible by 21. Insufficient.

2) The remainder when (m-n) is divided by 21 is 1.

Say m - n = 21b + 1; where b is any integer (quotient)

We cannot conclude on this basis whether (m^2 - n^2) is divisible by 21. Insufficient.

(1) and (2) combined:

From both the statements, we get that m^2 - n^2 = (21a + 1)*(21b + 1) = 21^2.ab + 21b + 21a + 1

Since 21^2.ab + 21b + 21a is divisible by 21, the remainder when 21^2.ab + 21b + 21a + 1 divided by 21 is 1. Sufficient.

Hope this helps!

-Jay
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