Question 50
Let $p$ be a prime number and let $0 \leq a_{1} \lt a_{2} \lt \cdots \lt a_{m} \lt p$ and $0 \leq b_{1} \lt b_{2} \lt \cdots \lt b_{n} \lt p$ be arbitrary integers. Denote by $k$ the number of different remainders of the numbers $a_{i} + b_{j}$, $1 \leq i \leq m$, $1 \leq j \leq n$, modulo $p$. Prove that:
a) if $m + n \gt p$, then $k = p$;
b) if $m + n \leq p$, then $k \geq m + n - 1$.
a) if $m + n \gt p$, then $k = p$;
b) if $m + n \leq p$, then $k \geq m + n - 1$.