Euler gone wild, one last time

\[\large a^{9074}\equiv{a^2}\pmod{19845}\]

How many integers \(a\) with \(1\leq{a}\leq1000\) satisfy the congruency above?

Extra Credit Question: How many integers \(a\) with \(1\leq{a}\leq1000\) satisfy the congruency \[\large a^{758}\equiv{a^2}\pmod{19845}\]

Inspiration here, here, and here.

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