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## Official Guide Explanation:Problem Solving #23

Background

This is just one of hundreds of free explanations I've created to the quantitative questions in The Official Guide for GMAT Review (12th ed.). Click the links on the question number, difficulty level, and categories to find explanations for other problems.

These are the same explanations that are featured in my "Guides to the Official Guide" PDF booklets. However, because of the limitations of HTML and cross-browser compatibility, some mathematical concepts, such as fractions and roots, do not display as clearly online.

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.)

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