Four bells toll at intervals of 6, 8, 12 and 18 minutes. If they toll together at 8:00 AM, when do they next toll together, and how many times in 12 hours?
LCM(6, 8, 12, 18) = 72 minutes, so they next ring together at 9:12 AM. In 12 hours (720 minutes) they ring together 720 / 72 = 10 more times, or 11 times counting the 8:00 AM ring.
Explain it simply
Each bell keeps its own beat, like four friends who visit the college canteen every 6, 8, 12 and 18 minutes. They meet again only at a moment that is a multiple of every interval. The first such moment is the least common multiple. Anything "happening together again" points to LCM; anything "cutting into equal largest pieces" points to HCF.
Example
Question: Bells toll at 6, 8, 12 and 18 minute intervals and start together at 8:00 AM.
- Prime factors: 6 = 2 x 3, 8 = 2^3, 12 = 2^2 x 3, 18 = 2 x 3^2.
- LCM = 2^3 x 3^2 = 72 minutes = 1 hour 12 minutes.
- Next together: 8:00 AM + 1:12 = 9:12 AM.
- In 12 hours = 720 minutes: 720 / 72 = 10 more times. Including the ring at 8:00, that is 11.
const g = (a, b) => (b === 0 ? a : g(b, a % b));
const l = [6, 8, 12, 18].reduce((a, b) => (a * b) / g(a, b));
console.log(l); // Output: 72
console.log(720 / l + 1); // Output: 11What to say in the interview
"They coincide at common multiples of every interval, so I need the LCM, 72 minutes. That puts the next joint ring at 9:12. In 12 hours there are 10 more joint rings after the first, 11 including it. I would confirm with the interviewer whether the starting ring counts."
Common mistakes
- Multiplying all intervals (6 x 8 x 12 x 18) instead of taking the LCM.
- Forgetting to say whether the starting ring is included in the count.