Question

(True/False): For every integer n, n is a multiple of 3 if and only if n can be expressed as the sum of 3 consecutive integers.

Answer 1

If n is a multiple of 3:

Let n = 3a, where a is an integer.

Then n - 1 + n + n + 1 = 3n = 3(3a) = 9a.

If n can be expressed as the sum of 3 consecutive integers:

n = a - 1 + a + a + 1 = 3a, so n is a multiple of 3.