Official Guide Explanation:
Problem Solving #23

Background

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Solution and Metadata

Question: 23
Page: 155
Difficulty: 5 (Moderate)
Category 1: Arithmetic > Properties of Integers > Remainder
Category 2: Arithmetic > Properties of Integers > Primes

Explanation: The easiest way to find an easier is to test a prime number greater than 3. If n = 5, n² = 25. When 25 is divided by 12, the quotient is 2 and the remainder is 1, choice (B).

To understand the concept more thoroughly, think about it this way. Any prime number greater than 3 is odd, which we can express as (even + 1). If we square it, the result is:

(e + 1)² = e² + 2e + 1

e² is always a multiple of four--multiply two evens together and you always get a multiple of 4.

2e is also, by the same principle, always a multiple of four. So, e² + 2e is always a multiple of 4.

Add one to a multiple of 4 and you have a number that, when divided by 4, has a remainder of 1.

That principle is occasionally applicable on other GMAT problems. Establishing that the remainder when divided by 12 is also 1 is more involved, and less applicable to other problems. (The explanation in The Official Guide itself does explain how, though I wouldn't worry about it.)

Click here for the full list of GMAT OG12 explanations.

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