Question

Question 2
Prove that one and only one out of n, (n + 2) and (n + 4) is divisible by 3, where n is any positive integer.

Answer 1

On dividing ‘n’ by 3, let q be the quotient and r be the remainder.
Then, n = 3q + r, where, r = 0, 1, 2
$\Rightarrow$ n = 3q or n = 3q + 1 or n = 3q + 2

Case I
If n = 3q, then n is only divisible by 3.
But n + 2 and n + 4 are not divisible by 3.

Case II
If n = 3q + 1, then (n + 2) = 3q + 3 = 3(q + 1) which is only divisible by 3.
But, n and n + 4 are not divisible by 3.
So, in this case, (n + 2) is divisible by 3.

Case III
When n = 3q + 2, then (n + 4) = 3q + 6 = 3(q + 2), which is only divisible by 3. But, n and (n + 2) are not divisible by 3.
Hence, one and only one out of n, (n + 2) and (n + 4) is divisible by 3.