Question

What strategies could you use to solve this problem? Santa rings his silver bell every 4 minutes and his golden bell every 6 minutes. He rang both bells at the same time just now.
In how many minutes will he ring both bells together again?

Answer 1

You need the least common multiple (LCM) of 4 and 6 — the smallest positive time that’s a multiple of both.

Strategies:

1. List multiples
- Multiples of 4: 4, 8, 12, 16, ...
- Multiples of 6: 6, 12, 18, ...
- First common multiple is 12 → answer: 12 minutes.

2. Prime factors
- 4 = 2^2, 6 = 2·3, take highest powers: 2^2·3 = 12.

3. Use gcd formula
- gcd(4,6) = 2, so lcm = (4·6)/2 = 12.

4. Congruences / timeline thinking
- Solve t ≡ 0 (mod 4) and t ≡ 0 (mod 6) → t is a multiple of lcm(4,6) = 12.

So Santa will ring both bells together again in 12 minutes.