Show that if $N = p^c$ for some prime $p$, then taking $a = p^c + 1 − p^{c−1}$, then $a^{N−1} \not≡ 1 \bmod N$ because $a^{N−1} ≡ p^{c−1} + 1 \bmod N$ and $p^{c−1} + 1 \not≡ 1\bmod N$
I am reading these notes on quantum computation and came across this exercise, but I'm unsure how to go about proving it. Any help is greatly appreciated as I have been fumbling around with some algebra to no avail for quite some time.