Question

Which of the following expressions is never a prime number when $p$ is a prime number?

$p^2 + 16$

$p^2 + 24$

$p^2 + 26$

$p^2 + 46$

$p^2 + 96$

Answer: C

Concepts:prime modular arithmetic

Difficulty rating: 1500

Small Hint:

Test the expressions modulo $3;$ for a prime $p \ne 3,$ $p^2 \equiv 1 \pmod 3$

Big Hint:

For the correct choice, also check $p = 3$ separately, then show it is a multiple of $3$ for all other primes

Solution:

Look at $p^2 + 26.$ When $p = 3,$ it’s $35 = 5 \cdot 7.$ For any other prime, $p$ isn’t divisible by $3,$ 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.

2018 AMC 10B Problem 11

Answer: C

Difficulty rating: 1500

Small Hint:

Big Hint:

Solution:

Problem 11 in Other Years