Square of a number coprime to 6 modulo 24

Statement

If an integer $m$ is divisible by neither $2$ nor $3$, then $m^2\equiv 1\pmod{24}$. Verify this by computing $35^2 \bmod 24$.

Solution

$35$ is divisible by neither $2$ nor $3$, so $\gcd(35,6)=1$. We have $35^2=1225=24\cdot 51+1,$ hence $35^2\equiv \boxed{1}\pmod{24}.$