### 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.

# Rational Round

### Why do this problem?

It provides an easy starter where all students ought to have success. It may seem surprising that some circles contain points with rational coordinates and others do not. The second half can be proved using modulus arithmetic and an argument by contradiction.

### Key Question

What if the circle$x^2 + y^2 = 3$ DID contain rational points...?

### Possible support

The article Modulus Arithmetic and a Solution to Dirisibly Yours gives a beginnersintroduction to modulus arithmetic and it is a good idea to try the problem Dirisibly Yours first.