### Double Time

Crack this code which depends on taking pairs of letters and using two simultaneous relations and modulus arithmetic to encode the message.

### 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?

# Modular Fractions

##### Age 16 to 18 Challenge Level:

$$3^{-1}+6^{-1} = 5 + 6 = 11 = 4 \pmod 7$$

and

$$(3+6)^{-1} = 9^{-1} = 2^{-1} = 4 \pmod 7$$

so $x=3,\ y=6$ is one solution. Now find the other solutions.