NL-NWO 2017 Problem 4

If we divide the number $13$ by the three numbers $5, 7$ , and $9$ , then these divisions leave remainders: when dividing by $5$ the remainder is $3$ , when dividing by $7$ the remainder is $6$ , and when dividing by $9$ the remainder is 4. If we add these remainders, we obtain $3 + 6 + 4 = 13$ , the original number. (a) Let $n$ be a positive integer and let $a$ and $b$ be two positive integers smaller than $n$ . Prove: if you divide $n$ by $a$ and $b$ , then the sum of the two remainders never equals $n$ . (b) Determine all integers $n > 229$ having the property that if you divide $n$ by $99, 132$ , and $229$ , the sum of the three remainders is $n$ .