Official Guide Explanation:
Problem Solving #144




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Solution and Metadata

Question: 144
Page: 81
Difficulty: 6 (Moderately Difficult)
Category 1: Algebra > Symbolism >
Category 2: Arithmetic > Properties of Integers > Primes

Explanation: The symbol is equivalent to factorial notation, so the question is interested in the range from (6! + 2) to (6! + 6). A prime number is an integer with no factors aside from 1 and itself, so if you can find another factor, you can show that a number is not prime. For instance:

(6! + 2) = 6(5)(4)(3)(2)(1) + 2 = 2((6)(5)(4)(3)(1) + 1) Since 2 is a factor of this number, it is not prime.

Note that the same manipulation can be done to any number between (6! + 2) to (6! + 6). As long as the constant is between 2 and 6, it can be factored out of both itself and the factorial. Therefore, there are no prime numbers in this range, choice (A).

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