Alliance Form 3 P1 Q2 — LCM & Time (Ringing Bells)

KCSE (Form 3) Form 3 Numbers

Published

The Question

“Three bells ring at intervals of 9, 15 and 21 minutes. They all ring together at 11:00 at night. Find the time they last rang together just before that.”

1

Understand what 'ringing together' means

The bells only chime at the same instant when the elapsed time is a multiple of every interval at once. The gap between one shared chime and the next is therefore the lowest common multiple of the three intervals, so finding that LCM is the whole problem.

2

Break each interval into prime factors

To find the LCM cleanly, split each number into primes. This lets you see exactly which prime factors are needed and how many of each.

9=3×3=329 = 3 \times 3 = 3^{2}
15=3×515 = 3 \times 5
21=3×721 = 3 \times 7
3

Build the LCM

Take each prime the greatest number of times it appears in any single number. The prime 3 appears twice in 9, so use 3 squared; then include one 5 and one 7. Multiplying these together gives the LCM in minutes.

LCM=32×5×7=315 minutes\text{LCM} = 3^{2} \times 5 \times 7 = 315 \text{ minutes}

So the three bells all ring together once every 315 minutes.

4

Convert 315 minutes into hours and minutes

A gap of 315 minutes is hard to picture on a clock, so change it to hours. Since 60 minutes make one hour, divide 315 by 60 to get the whole hours and the remaining minutes.

315÷60=5 remainder 15⇒5 h 15 min315 \div 60 = 5 \text{ remainder } 15 \Rightarrow 5 \text{ h } 15 \text{ min}
5

Count back from 11:00 p.m.

The bells rang together at 11:00 p.m., and the previous shared chime was 5 hours 15 minutes earlier. Using the 24-hour clock, 11:00 p.m. is 2300 hours; subtract the interval to find the earlier time.

2300−0515=17452300 - 0515 = 1745
1745=5:45 p.m.1745 = 5{:}45 \text{ p.m.}

Final Result

The bells last rang together at 5:45 p.m. (17:45 hours).

Why this method works

The method works because two events that repeat at fixed intervals only coincide when the elapsed time is a common multiple of both intervals; the first such coincidence after a shared start is the lowest common multiple. Prime factorisation guarantees the smallest such number because taking each prime to its highest power builds the smallest value divisible by all three intervals — any smaller value would be missing a needed factor. Once you know the bells coincide every 315 minutes, the moment before any known chime is simply that known time minus one full LCM interval.

✓

5 h 15 min before 11:00 p.m. is 5:45 p.m.; adding 315 minutes to 5:45 p.m. returns exactly to 11:00 p.m., confirming the interval.