The last digit of $2^{2006}$ My $13$ year old son was asked this question in a maths challenge. He correctly guessed $4$ on the assumption that the answer was likely to be the last digit of $2^6$. However is there a better explanation I can give him?

$2^{4} = 16$.  Multiply any even integer by $6$ and you don't change the last digit:
$0 \times 6 = 0$, $2 \times 6 = 12$, $4 \times 6 = 24$ etc.  The same is true if you multiply an even integer by anything whose last digit ends in $6$, in particular by $16$.  Now 
$2006 = 2004 + 2$ where $2004 = 501 \times 4$, so
$2^{2006} = (2^4)^{501} \times 2^2$ has the same last digit as $2^2$.
