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

Introduction

In this problem, we study the divisibility of a set of five numbers. We will show that among the numbers n, n+4, n+8, n+12 and n+16, exactly one number is divisible by 5, where n is any positive integer.

YouTube Video

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

Let n be any positive integer. Every positive integer can be written in one of the following five forms:

5q, 5q+1, 5q+2, 5q+3, or 5q+4, where q is an integer.

We now examine each case.

If n = 5q, then n is divisible by 5.

If n = 5q+1, then n+4 = 5q+5, which is divisible by 5.

If n = 5q+2, then n+8 = 5q+10, which is divisible by 5.

If n = 5q+3, then n+12 = 5q+15, which is divisible by 5.

If n = 5q+4, then n+16 = 5q+20, which is divisible by 5.

From all the above cases, we see that in each situation, exactly one number among n, n+4, n+8, n+12 and n+16 is divisible by 5.

Conclusion

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

Hence proved.