Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer.

Solution:
By Euclid’s division lemma, we have
a = bq + r; 0 ≤ r < b
For a = n and b = 5, we have
n = 5q + r …(i)
Where q is an integer
and 0 ≤ r < 5, i.e. r = 0, 1, 2, 3, 4.
Putting r = 0 in (i), we get
n = 5q
=> n is divisible by 5.
n + 4 = 5q + 4
=> n + 4 is not divisible by 5.
n + 8 = 5q + 8
=> n + 8 is not divisible by 5.
n + 12 = 5q + 12
=> n + 12 is not divisible by 5.
n + 16 = 5q + 16
=> n + 16 is not divisible by 5.
Putting r = 1 in (i), we get
n = 5q + 1
=> n is not divisible by 5.
n + 4 = 5q + 5 = 5(q + 1)
=> n + 4 is divisible by 5.
n + 8 = 5q + 9
=> n + 8 is not divisible by 5.
n + 12 = 5q + 13
=> n + 12 is not divisible by 5.
n + 16 = 5q + 17
=> n + 16 is not divisible by 5.
Putting r = 2 in (i), we get
n = 5q + 2
=> n is not divisible by 5.
n + 4 = 5q + 9
=> n + 4 is not divisible by 5.
n + 8 = 5q + 10 = 5(q + 2)
=> n + 8 is divisible by 5.
n + 12 = 5q + 14
=> n + 12 is not divisible by 5.
n + 16 = 5q + 18
=> n + 16 is not divisible by 5.
Putting r = 3 in (i), we get
n = 5q + 3
=> n is not divisible by 5.
n + 4 = 5q + 7
=> n + 4 is not divisible by 5.

n + 8 = 5q + 11
=> n + 8 is not divisible by 5.
n + 12 = 5q + 15 = 5(q + 3)
=> n + 12 is divisible by 5.
n + 16 = 5q + 19
=> n + 16 is not divisible by 5.
Putting r = 4 in (i), we get
n = 5q + 4
=> n is not divisible by 5.
n + 4 = 5q + 8
=> n + 4 is not divisible by 5.
n + 8 = 5q + 12
=> n + 8 is not divisible by 5.
n + 12 = 5q + 16
=> n + 12 is not divisible by 5.
n + 16 = 5q + 20 = 5(q + 4)
=> n + 16 is divisible by 5.
Thus for each value of r such that 0 ≤ r < 5 only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5.

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