Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

Which of these pocket money systems would you rather have?

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

What is the least square number which commences with six two's?

As I was going to St Ives, I met a man with seven wives. Every wife had seven sacks, every sack had seven cats, every cat had seven kittens. Kittens, cats, sacks and wives, how many were going to St. . . .

Can you find what the last two digits of the number $4^{1999}$ are?

Investigate $1^n + 19^n + 20^n + 51^n + 57^n + 80^n + 82^n $ and $2^n + 12^n + 31^n + 40^n + 69^n + 71^n + 85^n$ for different values of n.

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

What is the largest number you can make using the three digits 2, 3 and 4 in any way you like, using any operations you like? You can only use each digit once.