Square Modulo 5/Corollary

Theorem
When written in conventional base $10$ notation, no square number ever ends in one of $2, 3, 7, 8$.

Proof
The absence of $2$ and $3$ from the digit that can end a square follows directly from Square Modulo 5.

As $7 \equiv 2 \pmod 5$ and $8 \equiv 3 \pmod 5$, the result for $7$ and $8$ follows directly.