More Mods

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

Novemberish

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.

Rachel's Problem

Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!

A Biggy

Stage: 4 Challenge Level:

Find the smallest positive integer $N$ such that ${N\over 2}$ is a perfect cube, ${N\over 3}$ is a perfect fifth power and ${N\over 5}$ is a perfect seventh power.