GATE 2022 EE GA Q.3: solution and answer

Official answer: ASource: Official answer key, GATE 2022 EE (IIT Kharagpur)

1. All three bells ring together again after a time that is a multiple of every period. The first such time is the least common multiple (LCM) of $20$, $30$ and $50$ minutes.

2. Prime factorisation: $20=2^2\times 5$, $30=2\times 3\times 5$, $50=2\times 5^2$.

3. LCM rule: take the highest power of each prime that appears. $$\text{LCM}=2^2\times 3\times 5^2=4\times 3\times 25=300\text{ minutes}$$

4. Convert to hours: $300\text{ minutes}=\frac{300}{60}\text{ hours}=5\text{ hours}$.

5. Next time all three ring together: $12:00\text{ PM}+5\text{ hours}=5:00\text{ PM}$.

6. Check: $300/20=15$, $300/30=10$ and $300/50=6$ are all whole numbers. No smaller time works: the smaller multiples of $50$ are $50,100,150,200,250$, and each fails for another bell ($50$, $150$ and $250$ are not multiples of $20$; $100$ and $200$ are not multiples of $30$). So the answer is (A).