Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 1G (Number Theory)

A primitive root modulo $N$ is a unit of order $\phi (N)$; a nonzero quadratic residue modulo an odd prime $p$ is a square.

If a primitive root $g$ modulo $p$ were $x^2$, then its order would divide $(p-1)/2$, the order of the subgroup of squares, contradicting ${\mathrm{ord}}_p(g)=p-1$.

Modulo $11$, $2$ has order $10$, so the primitive roots are $2^k$ for $\gcd (k,10)=1$:Finally $2^{10}=1024\equiv 56\not\equiv 1(\mathrm{mod}121)$, so the lifting criterion makes $2$ primitive modulo $121$. It is plainly the smallest possible positive primitive root.