Exercise 2.1.8 (Unit digit of a square)

Proof. Let $a$ be the last digit of an integer $n$. The following array gives the square of $a$ modulo $10$.

Since $n\equiv a(\mathit{mod}10)$, $n^2\equiv a^2(\mathit{mod}120)$, so $n^2$ must have one of the following for its unit digit: $0,1,4,5,6,9$. □