We know that any positive integer is of the form 5q, 5q + 1 or 5q + 2, 5q + 3 or 5q + 4 for some integer q and one and onlyone of these possibilities can occur. So, we have the following cases:
Case-I: When n = 5q
In this case, we have
n = 5q, which is divisible by 5
Now, n = 5q
⇒ n + 4 = 5q + 4
⇒ n + 4 leaves remainder 4 when divided by 5
⇒ n + 4 is not divisible by 5.
Now n + 8 = 5q + 8 = 5(q + 1)+ 3 = 5m+3, m is an integer.
Clearly, n + 8 is not divisible by 5.
Again, n + 12 = 5q + 12 = 5(q + 2) + 2 = 5m + 2, m in an integer.
Clearly n + 12 is not divisible by 5.
Now n + 16 = 5q + 16 = 5(q + 13) + 1 = 5m + 1, m is an integer
⇒ n + 16 is not divisible by 5
Thus, if n = 5q only one out of n, n + 4, n + 8, n +
12 and n + 16 is divlsible by 5,
Similarly, this result can be proved for the rest of .

the cases.