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Adding in Rows

Age 11 to 14 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 + 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?