Lowest common multiple
Two lighthouses flash at different rates: one every $12$ seconds, the other every $16$. If they both flash together at $22{:}00$, when will they next flash together again? To answer this you need the lowest common multiple.
What is the L.C.M.?
The simplest way to find the L.C.M. (although a slow one) is to write out the multiples of each number until you see which one appears in every list first.
To start practicing, we are going to use the method of writing out the list of multiples. Write the first $10$ multiples of the numbers $4$ and $5$ (not counting zero).…
To start practicing, we are going to use the method of writing out the list of multiples. Write the first $10$ multiples of the numbers $4$ and $5$ (not counting zero). Then, identify which one is the least common multiple ($l.c.m.$) of both.
Look for the first number that appears in both lists.
Quickly calculate the $m.c.m. (3, 9)$. Look carefully at the relationship between the two numbers before you start calculating. Is one a multiple of the other? Explain…
Quickly calculate the $m.c.m. (3, 9)$. Look carefully at the relationship between the two numbers before you start calculating. Is one a multiple of the other? Explain whether you need to do many calculations in this case.
Finding the L.C.M. using prime factors
For large numbers, listing multiples is slow. It is quicker to break each number down into prime factors and combine them.
Calculate the least common multiple of the following pairs of numbers using prime factorization: a) $\text{m.c.m.}(10, 15)$ b) $\text{m.c.m.}(8, 12)$ c)…
Calculate the least common multiple of the following pairs of numbers using prime factorization:
- a) $\text{m.c.m.}(10, 15)$
- b) $\text{m.c.m.}(8, 12)$
- c) $\text{m.c.m.}(20, 30)$
To unlock a special level in a maths video game, you need to find the access code. The code is the lowest common multiple of the following three numbers: $36$, $48$ and…
To unlock a special level in a maths video game, you need to find the access code. The code is the lowest common multiple of the following three numbers: $36$, $48$ and $60$.
Find the prime factorisation of each number and calculate the final code.
"Coinciding events" problems
Almost all problems about lighthouses, buses, bells or lights that "go off together again" are solved with the L.C.M.: each rate gives its own multiples, and the next time they coincide is the L.C.M.
Three church bells ring at different intervals: the first rings every $6$ minutes, the second every $9$ minutes and the third every $12$ minutes. If all three rang…
Three church bells ring at different intervals: the first rings every $6$ minutes, the second every $9$ minutes and the third every $12$ minutes. If all three rang together at $10:00$ in the morning, at what time will they coincide again?
Two lights on a concert stage blink at different rates. The orange light turns on every $12$ seconds and the white light every $15$ seconds. If both lights have just…
Two lights on a concert stage blink at different rates. The orange light turns on every $12$ seconds and the white light every $15$ seconds. If both lights have just turned on at the same time at the start of a song:
- a) How many seconds will it take for them to be on together again for the first time?
- b) How many times will they have coincided during the first $3$ minutes of the song?
The figure shows the first times each light turns on.
Three school buses depart from the same central station at $08:00$ in the morning. The Route A bus stops at the station every $45$ minutes, the Route B bus every $60$…
Three school buses depart from the same central station at $08:00$ in the morning. The Route A bus stops at the station every $45$ minutes, the Route B bus every $60$ minutes, and the Route C bus every $90$ minutes.
At what time will all three buses coincide at the station again for the first time?
Back to the start: Since $12 = 2^2 \cdot 3$ and $16 = 2^4$, the $L.C.M.(12, 16) = 2^4 \cdot 3 = 48$ seconds; they will flash together again at $22{:}00{:}48$.