According to Euclid’s division Lemma,
Let the positive integer = n
And b=3
n =3q+r, where q is the quotient and r is the remainder
0<r<3 implies remainders may be 0, 1 and 2
Therefore, n may be in the form of 3q, 3q+1, 3q+2
When n=3q
n+2=3q+2
n+4=3q+4
Here n is only divisible by 3
When n = 3q+1
n+2=3q=3
n+4=3q+5
Here only n+2 is divisible by 3
When n=3q+2
n+2=3q+4
n+4=3q+2+4=3q+6
Here only n+4 is divisible by 3
So, we can conclude that one and only one out of n, n + 2 and n + 4 is divisible by 3.
Hence Proved