2018 AMC 10B Problem 11

Below is the professionally curated solution for Problem 11 of the 2018 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2018 AMC 10B solutions, or check the answer key.

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Concepts:prime modular arithmetic

Difficulty rating: 1500

11.

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

$p^2 + 16$

$p^2 + 24$

$p^2 + 26$

$p^2 + 46$

$p^2 + 96$

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.

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