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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 and 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?
Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!