Challenge Level

$6^1 = 6$

$6^2 = 36$

$6^3 = 216$

$6^4 = ...6$ since the last digit comes from multipying the previous last digit by $6$

Last digit of $66^{66}$ is $6$

Divide by $2$: Last digit will be $3$ or $8$

$66^{66}$ will be a multiple of $ 4$ (because $66$ and $66^{65}$ are both even) so $66^{66}\div2$ is even so the last digit must be $8$, not $3$.

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.