Lowest common multiple
Two lighthouses off the coast of Kerry flash at different rates: one every $12$ seconds, the other every $16$ seconds. If they both flash together at $10{:}00$ p.m., 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 LCM (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 LCM 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.
Find all natural numbers less than $150$ that are common multiples of both $4, 6$ and $9$. To do this: a) Calculate the $m.c.m.(4, 6, 9)$ first. b) Write the sequence of…
Find all natural numbers less than $150$ that are common multiples of both $4, 6$ and $9$. To do this:
- a) Calculate the $m.c.m.(4, 6, 9)$ first.
- b) Write the sequence of multiples of that value up to the given limit.
- c) What is the largest multiple of these three numbers that is less than $500$?
The least common multiple of two natural numbers, $24$ and $b$, is equal to $120$. That is, $\text{lcm}(24, b) = 120$. a) Find the prime factorisation of both the number…
The least common multiple of two natural numbers, $24$ and $b$, is equal to $120$. That is, $\text{lcm}(24, b) = 120$.
- a) Find the prime factorisation of both the number $24$ and the result, $120$.
- b) Deduce which prime factor must necessarily appear in the factorisation of $b$ and explain why no number lacking that prime factor could satisfy the condition.
- c) Find the three smallest values that the number $b$ can take, knowing that $b > 1$. Verify in each case that the $\text{lcm}(24, b)$ still results in exactly $120$.
- d) Could $b = 120$? Reason whether it is possible based on the divisibility relationship between $24$ and $120$.
"Coinciding events" problems
Almost all problems about lighthouses, buses, bells or lights that "go off together again" are solved with the LCM: each rate gives its own multiples, and the next time they coincide is the LCM.
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 athletes are training on a circular track. Aoife takes $60$ seconds to complete a full lap and Cian takes $75$ seconds. If they start together from the finish line,…
Two athletes are training on a circular track. Aoife takes $60$ seconds to complete a full lap and Cian takes $75$ seconds. If they start together from the finish line, how many seconds will it take for them to cross the finish line together again? How many laps will each have completed at that moment?
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?
Equal totals and fitting shapes together
The LCM also answers "smallest equal amount" questions. If footballs come in boxes of $6$ and rugby balls in boxes of $8$, the smallest number of each that can be equal is $\text{LCM}(6, 8) = 24$. That means $24 \div 6 = 4$ boxes of one and $24 \div 8 = 3$ boxes of the other.
It is also used when rectangles are laid side by side to make a square. If the rectangles all face the same way, the side of the square is made of whole widths in one direction and whole lengths in the other. So the side must be a multiple of both measurements, and the smallest square has side $\text{LCM}$.
For a solid cube made of bricks, do the same in all three directions: the edge is the LCM of the length, width and height of the brick, and the number of bricks is the volume of the cube divided by the volume of one brick.
On the shelf of a comic shop, they want to stack volumes of "Manga" that are $12$ mm thick and volumes of "Graphic Novel" that are $18$ mm thick. What is the minimum…
On the shelf of a comic shop, they want to stack volumes of "Manga" that are $12$ mm thick and volumes of "Graphic Novel" that are $18$ mm thick. What is the minimum height that both stacks must reach so that they are exactly the same height?
Aoife is organising a party and wants to buy hot dogs. The sausages come in packs of $6$ and the rolls in bags of $8$. What is the minimum number of hot dogs she can…
Aoife is organising a party and wants to buy hot dogs. The sausages come in packs of $6$ and the rolls in bags of $8$. What is the minimum number of hot dogs she can prepare so that no sausages or rolls are left over? How many packs of each product will she have to buy?
We want to cover the floor of a recording studio that is shaped like a perfect square with rectangular soundproofing panels $18$ cm wide by $24$ cm long. What must be…
We want to cover the floor of a recording studio that is shaped like a perfect square with rectangular soundproofing panels $18$ cm wide by $24$ cm long.
What must be the minimum length of the side of the room so that we can fit the panels without having to cut any? How many panels will we need in total to cover the entire floor?
We want to build a solid cube using toy bricks measuring $15$ cm long, $10$ cm wide and $6$ cm high. a) What must be the length of the edge of the smallest cube we can…
We want to build a solid cube using toy bricks measuring $15$ cm long, $10$ cm wide and $6$ cm high.
- a) What must be the length of the edge of the smallest cube we can form without cutting any bricks?
- b) How many toy bricks will we need in total to build this cube?
Back to the start: Since $12 = 2^2 \times 3$ and $16 = 2^4$, the $\text{LCM}(12, 16) = 2^4 \times 3 = 48$ seconds; they will flash together again at $10{:}00{:}48$ p.m.