2018 AMC 10B Problem 11

Problem 11 of 25IntermediateNumber Theory

Which of the following expressions is never a prime number when

p

is a prime number?

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Solution

Look at

p^2 + 26.

When

p = 3,

it’s

35 = 5 \cdot 7.

For any other prime,

p

isn’t divisible by

3,

p^2 \equiv 1 \pmod 3

and

p^2 + 26 \equiv 1 + 2 \equiv 0 \pmod 3.

Either way it’s a multiple of

3

bigger than

3,

hence composite. So it’s never prime. Thus, C is the correct answer.

Concepts: prime · modular arithmetic

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