Let n be any positive integer and b = 4

n =4q + r

where q is the quotient and r is the remainder

0 ≤ r < 4

So, the remainders may be 0, 1, 2 and 3

Hence, n may be in the form of 4q, 4q + 1, 4q + 2, 4q + 3

Case 1:

     If n = 4q + 1, then

=> n + 3= 4q + 3 + 1

=> n + 3 = 4q + 4

here n + 3 is only divisible by 4

Case 2:

     if n = 4q + 2

=> n + 6 = 4q + 6 + 2

=> n + 6 = 4q + 8

here only n + 6 is divisible by 4

Case 3:

     If n = 4q + 3

=> n + 9 = 4q + 3 + 9

=> n + 9 = 4q + 12

here only n + 9 is divisible by 12

Hence, it is clear that one and only one out of n, n + 3, n + 6 and n + 9 is divisible by 4