Prove this statement in complete sentence For every positive

Prove this statement in complete sentence For every positive integer, 3 divides n, or 3 divides n + d, or 3 divides n + 4

For every positive integer, there will be only 3 possible values of n mod(3) = (0,1,2)

Case 1:

n mod(3) = 0

In this case, n = 3k, therefore n is divisible by 3

Case 2:

n mod(3) = 2

In thie case, n=3k+2,

(n+4) = (3k+2) + 4 = 3k + 6

(3k+6)mod(3) = 0

Hence in this case (n+4) mod(3) will be equal to zero

Case 3:

n mod(3) = 1

In thie case, n=3k+1,

(n+2) = (3k+1) + 2 = 3k + 3

(3k+3)mod(3) = 0

Hence in this case (n+2) mod(3) will be equal to zero

Therefore for all the three cases, we can write either n is divisible by 3 or (n+2) is divisible by 3 or (n+4) is divisible by 3