Showing a polynomial has a solution in $ \mathbb{Z}/n\mathbb{Z} $

Show that the following equation has a solution in $ \mathbb{Z}/n\mathbb{Z} $ for every $n > 1$:

\begin{align} (x^2-2)(x^2-17)(x^2-34)=0 \end{align}

I know you can cite Chinese Remainder Theorem so it suffices to find solutions for powers of primes. Then you can somehow choose to consider only solutions $\pmod{2^e}$ * or $\pmod{8}$?* and $\pmod{p}$ for odd $p$. But I don't know how to get there or farther to solve the problem. I know the Legendre symbol can be used since $34 = 2 \cdot 17$.

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