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.
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 .