### 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?

### 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?

# Double Time

##### Age 16 to 18 Challenge Level:
To encode the following message the text was first written as pairs of letters ignoring the spaces between the words. For example John Brown would be written jo hn br ow nz where the z is added because there are an odd number of letters.

The letters of the alphabet are numbered $a=0,\ b=1,\ {\rm to}\ z=25$ and each pair of letters is replaced by a pair of numbers. There are $26\times 26=676$ pairs of numbers. It is possible to store the codes for all 676 pairs and look them up as needed but much more efficient to use a decyphering formula.

Each pair of numbers $(\alpha, \beta)$ is encoded as another pair of numbers $(\alpha',\beta')$ where $$\alpha' = \alpha + 3\beta \pmod {26}$$ $$\beta' = 5\beta \ \ \pmod{26}$$ Find $\alpha$ and $\beta$ in terms of $\alpha'$ and $\beta'$ and hence decode the following quotation which is a remark made by Einstein:

dj lb rn qm bu ao hd eo kr ia cs ud rx cm qo bn fr ld ek th ys wm