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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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Weekly Problem 3 - 2013
Can you find the value of t in these equations?

No Matter

Stage: 3 Short Challenge Level:
Your answer will be a multiple of $7$.

Let the number thought of be $x$. Then the final number is $$4(2x + 3) - 5- x = 7x + 7 = 7(x+ 1)$$

This problem is taken from the UKMT Mathematical Challenges.
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