Three bells toll at intervals of 9, 12 and 15 minutes respectively. If they start tolling together, after what time will they next toll together ?
Generated by claude-sonnet-4-6 · 2026-06-15 10:29 · grounding rag
Model Answer
The three bells will next toll together after LCM(9, 12, 15) minutes.
Prime factorisation:
- $9 = 3^2$
- $12 = 2^2 \times 3$
- $15 = 3 \times 5$
LCM $= 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = \mathbf{180}$ minutes
∴ The three bells will next toll together after 180 minutes (3 hours).
Source: Chapter 1, Section 1.2 – The Fundamental Theorem of Arithmetic
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Explanation
- The key concept is that the bells toll together at the LCM of their intervals (the smallest time when all three coincide again).
- Always show the prime factorisation step — examiners award marks for the working, not just the answer.
- LCM = product of the greatest powers of all prime factors involved.
- Converting 180 minutes to 3 hours is a good finishing touch but not compulsory.