OlympiadHardNumber Theory7–9

Three of Five Sum to a Multiple of Three

Três de Cinco com Soma Múltipla de Três

Prove that among any five whole numbers there are always three whose sum is divisible by $3$ .

Sort the numbers by their remainder on division by $3$ ; there are only three boxes, and two shapes of answer.

Solution

Step 1 of 4

1.Sort the five numbers into three boxes by remainder $0$ , $1$ or $2$ modulo $3$ .