Least Common Multiple (LCM)

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 whether you need to do many calculations in this case.

Calculate the least common multiple of the following pairs of numbers using prime factorization:

1. a) $\text{m.c.m.}(10, 15)$

1. b) $\text{m.c.m.}(8, 12)$

1. 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 $60$.

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 common multiple.

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. Lucía takes $60$ seconds to complete a full lap and Adrián 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 turned on at the same time at the start of a song:

1. a) How many seconds will it take for them to be on together again for the first time?

1. 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 $60$ 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?

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?

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?

A sports shop receives a delivery of footballs in boxes of $15$ units and basketballs in boxes of $20$ units. If the shop wants to have the same number of balls of each type for a promotion:

1. a) What is the minimum number of balls of each type it must have?
2. b) How many boxes of footballs and how many of basketballs does it need to buy to reach that quantity?

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.

1. a) What must be the length of the edge of the smallest cube we can form without cutting any bricks?
2. b) How many toy bricks will we need in total to build this cube?

An action figure collector has fewer than $1,000$ figures. He knows that if he organises them into boxes of $18$, none are left over. If he organises them into boxes of $20$, none are left over either, and if he groups them $24$ by $24$, the count is also exact.

1. a) What is the minimum number of figures he could have?
2. b) If he knows that the total number of figures is between $700$ and $800$, how many figures does he have exactly?

Three musicians in a jazz band are rehearsing a rhythmic pattern. The drummer hits the cymbal every $8$ beats, the bass player plays a low note every $12$ beats and the pianist plays a chord every $15$ beats.

1. a) If they start the bar together, on which beat will all three instruments play together again?
2. b) If the pianist decided to play every $18$ beats instead of $15$, would it take more or less time for them to coincide? Justify your answer with calculations.

An island lighthouse flashes a white light every $12$ seconds, and a nearby lighthouse flashes a red light every $16$ seconds. A third lighthouse flashes a green light every $20$ seconds. At $22:00$ hours, all three lights were seen to coincide.

1. a) Every how many minutes and seconds do all three lights coincide?
2. b) How many times in total will all three lights coincide between $22:00$ and $23:00$ hours? (Bear in mind that the first coincidence is at $22:00$).

Find all natural numbers less than $150$ that are common multiples of both $4, 6$ and $9$. To do this:

1. a) Calculate the $m.c.m.(4, 6, 9)$ first.
2. b) Write the sequence of multiples of that value up to the given limit.
3. c) What is the largest multiple of these three numbers that is less than $500$?