 Solution

The greatest common divisor issince $19$ and $44$ are coprime integers. Part (b) therefore gives $\boxed{1012/23=44}$ distinct $437$th powers in the nonzero multiplicative group of a finite field ${\mathbb{F}}_{1013}^{\times }$. If “powers mod $1013$” includes all residue classes, the additional value $0=0^{437}$ gives $\boxed{45}$ values including zero. The distinction is relevant because part (b) explicitly works in the nonzero group.