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If n is an integer, what is the remainder...

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by Vincen » Sun Sep 10, 2017 12:16 pm
If n is an integer, what is the remainder when (n - 3)^2 is divided by 4?
(1) n is divisible by 2.
(2) n is divisible by 4.

OA is D.

How can I solve this exercise? Is it a fast way to do it?
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Source: — Data Sufficiency |

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by Jay@ManhattanReview » Sun Sep 10, 2017 10:47 pm
Vincen wrote:If n is an integer, what is the remainder when (n - 3)^2 is divided by 4?
(1) n is divisible by 2.
(2) n is divisible by 4.

OA is D.

How can I solve this exercise? Is it a fast way to do it?
We have to determine the value of the remainder when (n - 3)^2 is divided by 4.

Statement 1: n is divisible by 2.

Say n = 2k; where k is an integer

Thus, (n - 3)^2 = (2k - 3)^2 = 4k^2 - 12k + 9

So, we have to determine the remainder when 4k^2 - 12k + 9 is divided by 4.

We see that 4k^2 is divisible by 4, leaving a remainder of 0; similarly, -12k is divisible by 4, leaving a remainder of 0; 9 is divisible by 4 leaves a remainder of 1.

Thus, the remainder = 1. Sufficient.

Sufficient.

Statement 2: n is divisible by 4.

Say n = 4k; where k is an integer

Thus, (n - 3)^2 = (4k - 3)^2 = 16k^2 - 24k + 9

With the same analysis as done in (1), we get the remainder = 1. Sufficient.

The correct answer: D

Hope this helps!

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by Jay@ManhattanReview » Sun Sep 10, 2017 10:59 pm
Vincen wrote:If n is an integer, what is the remainder when (n - 3)^2 is divided by 4?
(1) n is divisible by 2.
(2) n is divisible by 4.

OA is D.

How can I solve this exercise? Is it a fast way to do it?
There is a smarter way.

(n - 3)^2 = n^2 - 6n - 9 = n^2 - 3*(2n) - 9

If n is divisible by 2, n^2 and - 3*(2n) are divisible by 2, leaving a remainder of 0. And 9 divided by 4 leaves a remainder of 1.

Since Statement 1 and 2 say the same. Each of them itself is sufficient.

The correct answer: D

Hope this helps!

Download free ebook: Manhattan Review GMAT Quantitative Question Bank Guide

-Jay
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