### Purr-fection

What is the smallest perfect square that ends with the four digits 9009?

### Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

### Mod 7

Find the remainder when 3^{2001} is divided by 7.

# Dirisibly Yours

### Why do this problem?

An exercise in proof by induction or, perhaps more simply, modulus arithmetic.

### Possible approach

The challenge in the question is to find the neatest and simplest proof. The class could take up this challenge, perhaps working in pairs. Then the class could discuss criteria for a good write-up of a proof and the teacher can add advice. Perhaps the class could then mark each others work say in groups of four.

### Key questions

How couldwe test that an expression is divisible by 33?

What do we notice for small values of n?

What methods do we know for proving a result for all positive integers?

### Possible support

See the article related to this problem