sum of remainders of n divided by 99,132, 229 is n (n>229)
Source: Dutch NMO 2017 p4
September 7, 2019
remainderSumnumber theory
Problem Statement
If we divide the number by the three numbers , and , then these divisions leave remainders: when dividing by the remainder is , when dividing by the remainder is , and when dividing by the remainder is 4. If we add these remainders, we obtain , the original number.
(a) Let be a positive integer and let and be two positive integers smaller than . Prove: if you divide by and , then the sum of the two remainders never equals .
(b) Determine all integers having the property that if you divide by , and , the sum of the three remainders is .