Problem 6 · 2022 Math Kangaroo
Medium
Number Theory
arithmetic-sequenceoff-by-one
Kengu hops along the number line. He starts at 0, always makes two big jumps followed by three small jumps (see diagram), and keeps repeating this pattern. On which of these numbers will he land?
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Answer: C — 84
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Hint 1 of 2
One full cycle is two big jumps then three small jumps; find how far one cycle moves him and which spots he lands on.
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Hint 2 of 2
List the landing positions within one cycle, then see which cycle the answer lands in by working modulo the cycle length.
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Approach: find the repeating pattern of landing spots
- From the diagram one cycle is two big jumps of 3 then three small jumps of 1, so each full cycle moves him forward 6 + 3 = 9.
- Starting a cycle at a multiple of 9, he lands on +3, +6, +7, +8, +9 within it, so every landing spot is a multiple of 9 plus 3, 6, 7, 8 or 9.
- Since 84 = 81 + 3 = (9 x 9) + 3, it is a landing spot, while 82, 83, 85, 86 are not, so the answer is C.
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