The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time will they change together next ?
Generated by claude-sonnet-4-6 · 2026-06-15 10:30 · grounding rag
Model Answer
The lights will next change together after LCM(48, 72, 108) seconds.
Prime factorisation:
- $48 = 2^4 \times 3$
- $72 = 2^3 \times 3^2$
- $108 = 2^2 \times 3^3$
LCM = $2^4 \times 3^3 = 16 \times 27 = \mathbf{432}$ seconds
Converting: $432 \div 60 = 7$ minutes $12$ seconds
They started at 7:00:00 a.m., so they will next change together at 7 minutes 12 seconds after 7 a.m., i.e., at 7:07:12 a.m.
Source: Chapter 1, Section 1.2 – The Fundamental Theorem of Arithmetic
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Explanation
- The key insight: lights changing "together" means finding the LCM (smallest time that is a multiple of all three intervals).
- Use prime factorisation for LCM: take the highest power of each prime factor present.
- Always convert seconds to minutes and seconds for the final answer, and state the actual time — examiners expect this last step.
- Show factorisation clearly; each step carries marks.