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Problem 598

AMC 12 late, AIME early

Number theory Difficulty 5.0 Find the answer

1. [2] Find the number of positive integers $x$ less than 100 for which
$3^{x}+5^{x}+7^{x}+11^{x}+13^{x}+17^{x}+19^{x}$
is prime.

Your answer

Official solution

Answer: 0 We claim that our integer is divisible by 3 for all positive integers $x$ . Indeed, we have
$\begin{aligned} 3^{x}+5^{x}+7^{x}+11^{x}+13^{x}+17^{x}+19^{x} & \equiv(0)^{x}+(-1)^{x}+(1)^{x}+(-1)^{x}+(1)^{x}+(-1)^{x}+(1)^{x} \\ & \equiv 3\left[(1)^{x}+(-1)^{x}\right] \\ & \equiv 0 \quad(\bmod 3) . \end{aligned}$

It is clear that for all $x \geq 1$ , our integer is strictly greater than 3 , so it will always be composite, making our answer 0 .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.