Challenge Level

List any 3 numbers.

It is always possible to find a subset of adjacent numbers that add up to a **multiple of 3** (that is either one, two or three numbers that are next to each other). For example:

5, 7, 1

5 and 7 are adjacent andÂ 5 + 7 = 12 (a multiple of 3)

4, 4, 15

15 is a multiple of 3

5, 11, 2

5 + 11 + 2 = 18 (a multiple of 3)

Can you explain why and prove it?

What happens if you write a list of 4 numbers?

Is it always possible to find a subset of adjacent numbers that add up to a multiple of 4?

Can you explain why and prove it?

What happens if you write a long list of numbers (say n numbers)?

Is it always possible to find a subset of adjacent numbers that add up to a multiple of $n$?

Can you explain why and prove it?