In a teachers' workshop, the number of teachers teaching French, Hindi and English are 48, 80 and 144 respectively. Find the minimum number of rooms required if in each room the same number of teachers are seated and all of them are of the same subject.
Generated by claude-sonnet-4-6 · 2026-06-15 10:32 · grounding rag
Model Answer
To find the minimum number of rooms, we need the HCF of 48, 80 and 144 (maximum teachers per room, same subject).
Prime factorisation:
- $48 = 2^4 \times 3$
- $80 = 2^4 \times 5$
- $144 = 2^4 \times 3^2$
HCF(48, 80, 144) = $2^4$ = 16
So, 16 teachers can be seated in each room.
Number of rooms required:
- French: $48 \div 16 = 3$
- Hindi: $80 \div 16 = 5$
- English: $144 \div 16 = 9$
Total minimum rooms = 3 + 5 + 9 = 17
Source: Chapter 1, Section 1.2 – The Fundamental Theorem of Arithmetic
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Explanation
- The key insight is that the number of teachers per room must divide all three numbers exactly → this is the HCF.
- HCF gives the maximum number per room, which gives the minimum number of rooms — examiners specifically check this logic.
- Show prime factorisation clearly; take the lowest power of common prime factors for HCF.
- Don't forget to add the rooms for all three subjects at the end — that single addition step carries a mark.