Past exam of the mathematics course of the University of Cambridge / 2014 / ia / Paper 4 / 6E / iv / Solution

Fermat's little theorem reduces the exponent modulo $22$ because $23$ is a prime number and $5$ is coprime to it. Compute $5^2\equiv 3(\mathrm{mod}22)$, then $5^4\equiv 9(\mathrm{mod}22)$ and $5^5\equiv 45\equiv 1(\mathrm{mod}22)$. Thus $5^5=22k+1$ for an integer $k$, andThe exponent is $5^5=3125$, rather than an iterated left-associated power.