Let $ a $ be a positive integer and $ p $ be an odd prime. Show that if $ x^2 \equiv 1 \mod{p^a}, $ then $ x \equiv \pm 1 \mod{p^a}. $

Question

Let $ a $ be a positive integer and $ p $ be an odd prime. Show that if $ x^2 \equiv 1 \mod{p^a}, $ then $ x \equiv \pm 1 \mod{p^a}. $

From $ x^2 \equiv 1 \mod{p^a} $ we have that $ p^a \vert (x^2 - 1) = (x + 1)(x - 1). $ It's easy to see that the proof is true when $ a = 1 $ from Euclid's lemma.

How can I show this is true for $ a = 2. $ I am trying to build towards an induction proof.

Hints only, please.