Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 1G (Number Theory)

Since $136=8\cdot 17$ and the factors are coprime,Hence

The required integer is the exponent of the unit group. Since $\lambda (8)=2$ and $\lambda (17)=16$,

The residue $3$ is a primitive root modulo $17$. ChooseWriting $g=1+8k$ gives $k\equiv 13(\mathrm{mod}17)$, so one may takeIts Chinese-remainder components have orders $1$ and $16$, hence its order modulo $136$ is $16$.