Least Common Multiple
Two lighthouses flash at different rates: one every $12$ seconds and the other every $16$. If they both flashed together at $10{:}00$ p.m., when will they next flash together? To answer this you need the least common multiple.
What the LCM is
The simplest way (although slow) to find the LCM is to write out the multiples of each number until you see which one repeats 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 with prime factors
For large numbers, listing multiples is slow. It is faster to break the numbers 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 math video game, you need to find the access code. The code is the least common multiple of the following three numbers: $36$, $48$ and…
To unlock a special level in a math video game, you need to find the access code. The code is the least common multiple of the following three numbers: $36$, $48$ and $60$.
Find the prime factorization of each number and calculate the final code.
Find a number between $300$ and $400$ that is simultaneously a multiple of $12$, $15$ and $18$. Explain the steps you followed to find it using the concept of lowest…
Find a number between $300$ and $400$ that is simultaneously a multiple of $12$, $15$ and $18$. Explain the steps you followed to find it using the concept of lowest common multiple.
Problems about coinciding events and equal measures
Problems with lighthouses, trucks, bells, or lights that "go off together again" are solved with the LCM: each rate marks its own multiples, and the first time they coincide is the LCM.
The LCM is also useful when you need to put together equal amounts using pieces of different sizes. For example: hot dogs in packs of $6$ and buns in bags of $8$. So that nothing is left over, the total must be a multiple of $6$ and of $8$, and the smallest is $\text{LCM}(6, 8) = 24$. Then, the number of packs comes from dividing: $24 \div 6 = 4$ packs and $24 \div 8 = 3$ bags.
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?
Marta 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…
Marta 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?
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.
Back to the beginning: since $12 = 2^2 \times 3$ and $16 = 2^4$, $\text{LCM}(12, 16) = 2^4 \times 3 = 48$ seconds. The lighthouses will flash together again at $10{:}00{:}48$ p.m.