### More Mods

What is the units digit for the number 123^(456) ?

### N000ughty Thoughts

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the number of noughts in 10 000! and 100 000! or even 1 000 000!

### Mod 3

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

# Novemberish

##### Stage: 4 Challenge Level:

(a) A four digit number (in base $10$) $aabb$ is a perfect square. Discuss ways of systematically finding this number.
(b) Prove that $11^{10}-1$ is divisible by $100$.