According to Euclid’s division Lemma,
Let the positive integer = n
And, b=5
n = 5q+r, where q is the quotient and r is the remainder
0 < r < 5 implies remainders may be 0, 1, 2, 3, 4 and 5
Therefore, n may be in the form of 5q, 5q+1, 5q+2, 5q+3, 5q+4
So, this gives us the following cases:
CASE 1:
When, n = 5q
n+4 = 5q+4
n+8 = 5q+8
n+12 = 5q+12
n+16 = 5q+16
Here, n is only divisible by 5
CASE 2:
When, n = 5q+1
n+4 = 5q+5 = 5(q+1)
n+8 = 5q+9
n+12 = 5q+13
n+16 = 5q+17
Here, n + 4 is only divisible by 5
CASE 3:
When, n = 5q+2
n+4 = 5q+6
n+8 = 5q+10 = 5(q+2)
n+12 = 5q+14
n+16 = 5q+18
Here, n + 8 is only divisible by 5
CASE 4:
When, n = 5q+3
n+4 = 5q+7
n+8 = 5q+11
n+12 = 5q+15 = 5(q+3)
n+16 = 5q+19
Here, n + 12 is only divisible by 5
CASE 5:
When, n = 5q+4
n+4 = 5q+8
n+8 = 5q+12
n+12 = 5q+16
n+16 = 5q+20 = 5(q+4)
Here, n + 16 is only divisible by 5
So, we can conclude that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5.
Hence Proved
