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.
Question:

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:

Consider the numbers $n,(n+4),(n+8),(n+12)$ and $(n+16)$, where $n$ is any positive integer.

Suppose $n=5 q+r$, where $0 \leq r<5$

$n=5 q, 5 q+1,5 q+2,5 q+3,5 q+4$

(By Euclid’s division algorithm)

Case: 1

When n=5q.

n=5q is divisible by 5.n=5q is divisible by 5.

$n=5 q$ is divisible by 5

$n+4=5 q+4$ is not divisible by 5

$n+8=5 q+5+5+3=5(q+1)+3$ is not divisible by 5 .

$n+12=5 q+10+2=5(q+2)+2$ is not divisible by 5 .

$n+16=5 q+15+1=5(q+3)+1$ is not divisible by 5 .

Case: 2

When n=5q+1.

$n=5 q+1$ is not divisible by 5

$n+4=5 q+1+4=5(q+1)$ is divisible by 5

$n+8=5 q+1+5+3=5(q+1)+4$ is not divisible by 5 .

$n+12=5 q+1+12=5(q+2)+3$ is not divisible by 5

$n+16=5 q+1+16=5(q+3)+2$ is not divisible by 5

Case: 3

When n=5q+2.

$n=5 q+2$ is not divisible by 5 .

$n+4=5 q+2+4=5(q+1)+1$ is not divisible by 5 .

$n+8=5 q+2+8=5(q+2)$ is divisible by 5

$n+12=5 q+2+12=5(q+2)+4$ is not divisible by $5 .$

$n+16=5 q+2+16=5(q+3)+3$ is not divisible by 5 .

Case: 4

When n=5q+3.

$n=5 q+3$ is not divisible by 5

$n+4=5 q+3+4=5(q+1)+2$ is not divisible by 5 .

$n+8=5 q+3+8=5(q+2)+1$ is not divisible by 5 .

$n+12=5 q+3+12=5(q+3)$ is divisible by 5 .

$n+16=5 q+3+16=5(q+3)+4$ is not divisible by 5

Case: 5

When n=5q+4.

$n=5 q+4$ is not divisible by 5 .

$n+4=5 q+4+4=5(q+1)+3$ is not divisible by 5 .

$n+8=5 q+4+8=5(q+2)+2$ is not divisible by 5 .

$n+12=5 q+4+12=5(q+3)+1$ is not divisible by 5 .

$n+16=5 q+4+16=5(q+4)$ is divisible by 5

Hence, in each case, one and only one out of $n, n+4, n+8, n+12$ and $n+16$ is divisible by 5 .