Let three consecutive even numbers be : 2n, 2n + 2, 2n + 4
Their sum = 2n + 2n + 2 + 2n + 4
⇒ (2n + 2n + 2n) + (2 + 4)
⇒ 6n + 6 = 6(n + 1) = 6k, where k = n + 1.
Clearly 6k is divisible by 6.
Hence, proved that the sum of three consecutive even numbers is divisible by 6.