Problem 4 · 2018 Math Kangaroo
Easy
Number Theory
divisibility
Thor has seven stones and a hammer. With his hammer he hits a stone and it breaks into five small stones. He does that a few times. Which of these numbers could be the number of stones he ends up with?
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Answer: D — 23
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Hint 1 of 2
Each hammer blow replaces one stone with five, so the count changes by a fixed amount.
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Hint 2 of 2
Track the total as a starting value plus a multiple of that step.
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Approach: find the invariant: each hit adds a constant number of stones
- Breaking one stone into five removes 1 and adds 5, a net gain of +4 stones per hit.
- Starting at 7, after k hits the total is 7+4k: 11, 15, 19, 23, ...
- Among the options only 23 has the form 7+4k (k=4).
- So he can end with 23 stones.
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