### Double Time

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

### Modular Fractions

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

### Purr-fection

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

# Rational Round

##### Age 16 to 18Challenge Level

Show that for every integer $k$ the point $(x, y)$, where

$$x = {2k\over k^2 + 1}, \ y = {k^2 - 1\over k^2 + 1},$$

lies on the unit circle, $x^2 + y^2 =1$. That is, there are infinitely many rational points on this circle.

Show that there are no rational points on the circle $x^2 + y^2 =3$.