Question 24
Ats and Pets both thought of two positive integers that do not exceed some positive integer $n$. If they both added the numbers they thought of, then both sums gave the same remainder when divided by $n$. But if both of them multiplied the numbers they thought of, then both products also gave equal remainders when divided by $n$. Is it necessarily true that the numbers they thought of were the same, if
a) $n = 99$?
b) $n = 101$?
a) $n = 99$?
b) $n = 101$?