We'll pick up from the last challenge and imagine we've picked a modulus $p$, and we will restrict ourselves to the case when $p$ is prime.

The integers modulo $p$ define a field, denoted $\Fp$.

If the modulus is not prime, the set of integers modulo $n$ define a ring.

A finite field $\Fp$ is the set of integers ${0,1,...,p-1}$, and under both addition and multiplication there are inverse elements $b_{+}$ and $b_{*}$ for every element $a$ in the set, such that $a + b_{+} = 0$ and $a \cdot b_{*} = 1$.

Note that the identity element for addition and multiplication is different! This is because the identity when acted with the operator should do nothing: $a + 0 = a$ and $a \cdot 1 = a$.

Lets say we pick $p = 17$. Calculate $3^{17} \mod 17$. Now do the same but with $5^{17} \mod 17$.

What would you expect to get for $7^{16} \mod 17$? Try calculating that.

This interesting fact is known as Fermat's little theorem. We'll be needing this (and its generalisations) when we look at RSA cryptography.

Now take the prime $p = 65537$. Calculate $273246787654^{65536} \mod 65537$.

Did you need a calculator?