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What is the remainder when a is divided by 4?

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by BTGmoderatorDC » Wed Jul 10, 2019 10:38 pm

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Difficulty—

What is the remainder when a is divided by 4?

(1) a is the square of an odd integer.
(2) a is a multiple of 3.


OA A

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Source: — Data Sufficiency |

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by Jay@ManhattanReview » Thu Jul 11, 2019 5:27 am
BTGmoderatorDC wrote:What is the remainder when a is divided by 4?

(1) a is the square of an odd integer.
(2) a is a multiple of 3.

OA A

Source: Manhattan Prep
Let's take each statement one by one.

(1) a is the square of an odd integer.

The squares of first odd numbers are 1^2 = 1; 3^2 = 9; 5^2 = 25; 7^2 = 49

The numbers are 1, 9, 25, 49, 81 and so on. Each number divided by 4 leaves a remainder 1. Sufficient.

(2) a is a multiple of 3.

Few multiples of 3 are 3, 6, 9, 12...

While 3 divided 4 leaves remainder 3, 6 divided 4 leaves remainder 2. No unique answer. Insufficient.

The correct answer: A

Hope this helps!

-Jay
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by GMATGuruNY » Thu Jul 11, 2019 9:38 am
BTGmoderatorDC wrote:What is the remainder when a is divided by 4?

(1) a is the square of an odd integer.
(2) a is a multiple of 3.
Statement 1:
An odd integer can be written as follows:
2x+1, where x is an integer.
Since a is the square of an odd integer, we get:
a = (2x+1)² = (2x)² + 2(2x)(1) + 1² = 4x² + 4x + 1 = 4(x²+x) + 1 = (multiple of 4) + 1
Since a is equal to 1 more than a multiple of 4, dividing a by 4 will yield a remainder of 1.
SUFFICIENT.

Statement 2:
Case 1: a=6, with the result that a/4 = 6/4 = 1 R2
Case 2: a=9, with the result that a/4 = 9/4 = 2 R1
Since the remainder can be different values, INSUFFICIENT.

The correct answer is A.
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