### 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.

### A Biggy

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

# Negative Power

##### Stage: 4 Challenge Level:

This problem is a good one with which to illustrate the precision of meaning that mathematics often requires - in this case the precedence of the operations.

Superficially the task provides some practice at interpreting negative indices but more deeply it invites students to find a simplification for the 'stacked' indices.

The real variable each time is the value, of the four, which is selected as base. To explore that students might go on to consider which sets of $4$ values will produce the same 'stacked' value, irrespective of stacking order.