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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. Can you explain why and prove it?

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No Matter

Weekly Problem 2 - 2013
After performing some operations, what number is your answer always a multiple of?

Multiple Magic

Stage: 3 Short Challenge Level: Challenge Level:1

Call your first number $x$.

Doubling and adding five gives $2x+5$.

Doubling and adding two gives $2(2x+5)+2 = 4x+12$.

Subtracting $x$ then leaves $3x+12 = 3(x+4)$ which is always a multiple of 3.

This problem is taken from the UKMT Mathematical Challenges.