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## Official Guide Explanation:

Data Sufficiency #90

**Background**

This is just one of hundreds of free explanations I've created to the quantitative questions in The Official Guide for GMAT Quantitative Review (2nd ed.). Click the links on the question number, difficulty level, and categories to find explanations for other problems.

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**Solution and Metadata**

**Question****: 90**

Page: 159

Difficulty: **5** (Moderate)

Category 1: Arithmetic > Properties of Integers > Primes

Category 2: Arithmetic > Properties of Integers > Evens and Odds

**Explanation:** Statement (1) is insufficient. If n = 4, n cannot be written as the sum of two different positive primes. (2 + 2: not different; 3 + 1: 1 isn't prime.) However, if n = 5, n can be written as the sum of two different positive primes, 2 and 3.

Statement (2) is also insufficient: if n = 1, it cannot be written as the sum of two different positive numbers, let alone positive primes. However, if n = 5, n can be written as the sum of two different positive primes.

Taken together, the statements are still insufficient. If n = 5, n can be written as the sum of two different positive primes, but if n = 11, n cannot. To find this counterexample, look for the sum of 2 and a non - prime odd number; the only way an odd number can be the sum of two primes is if it is the sum of an even prime (2) and an odd prime. 11 is the sum of 2 and 9, and since 9 is not a prime, it cannot be the sum of two primes. Choice (E) is correct.

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