2018 AMC 10B Problem 11

Question

Which of the following expressions is never a prime number when

p

is a prime number?

p^2 + 16

p^2 + 24

p^2 + 26

p^2 + 46

p^2 + 96

Test the expressions modulo

3;

for a prime

p \ne 3,

p^2 \equiv 1 \pmod 3

For the correct choice, also check

p = 3

separately, then show it is a multiple of

3

for all other primes

Look at

p^2 + 26.

When

p = 3,

it’s

35 = 5 \cdot 7.

For any other prime,

p

isn’t divisible by

3,

so

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.

Problem 11 in other years