An expert question on cycles for P5–P6, with a step-by-step solution, diagrams and the answer. Cycle questions appear in the Hua Xia Cup from the P1 first round to the P5 papers: which comes in a given place, which day of the week, which last digit.
The question
Ming and May write numbers side by side, one at a time and at the same pace.
When they write the same number at the same moment for the 390th time, how many numbers has each written?
Show the hints (3)
💡Hint 1: After how many turns do they both start again together (Ming writing 1 and May writing 8)?
💡Hint 2: Every 24 turns is one big cycle. List both rows side by side for one big cycle: on which turns do they match, and how many per cycle?
💡Hint 3: One big cycle has 3 matches. 390 ÷ 3 = 130 exactly — work out which big cycle the 390th match falls in.
Worked solution
Have a go first, then read the solution step by step.
1.The lowest common multiple of 3 and 8 is 24: every 24 turns Ming is back at 1 and May at 8 — one big cycle.
2.Listing one big cycle of 24 turns, they match on turn 6 (3), turn 16 (1), turn 23 (2): 3.
3.390 ÷ 3 = 130 exactly: the 390th match is the last match of big cycle 130, turn 23 of that cycle.
4.So each has written 129 × 24 + 23 = 3119 numbers (both writing 2).
⚠️Common mistake: with an exact division, putting the 390th match in big cycle 131 and answering 3143; or assuming May also counts up from 1, so matches come first in each cycle.
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