Question

Prove that one out of every three consecutive integers is a multiple of three.

(3 Marks)

Answer 1

Let 3 consecutive positive integers be n, n+1 and n+2
Whenever a number is divided by 3, the remainder we get is either 0, or 1, or 2.
:
Therefore:

n = 3p or 3p+1 or 3p+2, where p is some integer
If n = 3p, then n is divisible by 3
If n = 3p+1, then n+2 = 3p+1+2 = 3p+3 = 3(p+1) is divisible by 3
If n = 3p+2, then n+1 = 3p+2+1 = 3p+3 = 3(p+1) is divisible by 3
Thus, we can state that one of the numbers among n, n+1 and n+2 is always divisible by 3