Problem 22 · 1989 AJHSME
Stretch
Number Theory
lcm-of-cycle-lengths
The letters A, J, H, S, M, E and the digits 1, 9, 8, 9 are "cycled" separately as follows and put together in a numbered list:
AJHSME 1989
1. JHSMEA 9891
2. HSMEAJ 8919
3. SMEAJH 9198
.........What is the number of the line on which AJHSME 1989 will appear for the first time?
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Answer: C — 12.
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Hint 1 of 3
The letters and the digits cycle on their own clocks: 6 letters need 6 steps to come home, 4 digits need 4 steps. When do BOTH come home at once?
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Hint 2 of 3
Two repeating cycles realign for the first time at the least common multiple of their lengths — the smallest number that's a whole count of each.
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Hint 3 of 3
List multiples of 6 (6, 12, 18, …) and of 4 (4, 8, 12, …) and catch the first they share.
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Approach: least common multiple of the two cycle lengths
- Each part is back to its start on its own schedule: the 6 letters realign every 6 lines, the 4 digits every 4 lines. The full word AJHSME 1989 reappears only when both happen on the same line.
- That first shared moment is the least common multiple: multiples of 6 are 6, 12, 18…; of 4 are 4, 8, 12…; the first match is LCM(6, 4) = 12.
- Trap to avoid: it is NOT 6 × 4 = 24. Two cycles can sync up sooner than their product whenever the lengths share a factor (here both share a 2), so always use the LCM, not the plain product. Why this transfers: any 'when do two repeating things coincide' question — gears, blinking lights, planet alignments — is an LCM.
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