Least Common Multiple

To start practicing, we are going to use the method of writing the list of multiples. Write the first $10$ multiples of the numbers $4$ and $5$ (not counting zero). Then, identify which is the least common multiple ($m.c.m.$) of both.

Work out the $m.c.m. (3, 9)$ quickly. Look closely 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) $m.c.m. (10, 15)$
2. b) $m.c.m. (8, 12)$
3. c) $m.c.m. (20, 30)$

To unlock a special level in a mathematical 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.

A gamer has the LED lights on their keyboard configured as follows: the red lights flash every $2$ seconds and the white lights flash every $3$ seconds. If they have just flashed at the same time right now, in how many seconds will they both flash together again for the first time?

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 have rung together at $10:00$ in the morning, at what time will they coincide again?

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

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

Two athletes are training on a circular track. Lucy takes $60$ seconds to complete one lap and Andrew takes $75$ seconds. If they start together from the finish line, how many seconds will it take them to cross the finish line at the same time again? How many laps will each have completed at that moment?

On a circular athletics track, Lucy completes a lap in $75$ seconds, while Marcus takes $90$ seconds. If they both start together from the finish line:

1. a) After how much time will they cross the finish line at the same time again? Express the result in minutes.
2. b) How many laps will each have completed at that moment?

On a go-kart track, two drivers, Lucy and Marcus, cross the finish line at the same time. Lucy takes $105$ seconds to complete a lap, while Marcus takes $90$ seconds.

1. a) How much time will pass until they cross the finish line together again for the first time? Express the result in minutes and seconds.
2. b) At that moment, how many laps will each driver have completed?
3. c) How many laps ahead will the faster driver be compared to the slower one?

Three school buses leave the same central station at $08:00$ in the morning. The Route A bus passes through the station every $45$ minutes, the Route B bus every $60$ minutes and the Route C bus every $90$ minutes.

At what time will the three buses coincide at the station again for the first time?

An influencer is planning their posts. They want to upload a Reel every $8$ days and a photo to their profile every $10$ days. Today, Monday, they have posted both.

In how many days will the posting of a Reel and a photo coincide again for the first time? If today is Monday, what day of the week will it be when they coincide again?

Three smart devices on a home network perform cloud backups at different frequencies. The first one does it every $45$ minutes, the second every $72$ minutes and the third every $2$ hours. If all three coincided in performing the backup at $10:00$ in the morning:

1. a) How many hours must pass at a minimum for them to coincide again?
2. b) At what exact time in the afternoon or evening will this new coincidence occur?
3. c) How many backups will each device have performed in that time interval?

Two lights on a concert stage flash 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 coincide turned on again for the first time?
2. b) How many times will they have coincided during the first $3$ minutes of the song?

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

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

A logistics company has three trucks that leave the depot for different cross-country routes. The first leaves every $12$ days, the second every $15$ days and the third every $18$ days. On the $1$ of January, all three coincided at the base.

1. a) Will the three trucks coincide at the base again before the end of June? Justify your answer by calculating the exact day of the coincidence.
2. b) How many times will each truck have left on a trip by that day?

On the shelf of a comic shop, they want to stack "Manga" volumes that are $12$ in thick and "Graphic Novel" volumes that are $18$ in thick. What is the minimum height both stacks must reach so that they measure exactly the same?

Emma is organizing a party and wants to buy hot dogs. The sausages come in packs of $6$ units and the buns in bags of $8$ units. What is the minimum number of hot dogs she can prepare so that there is not a single sausage or bun left over? How many packs of each product will she have to buy?

A sports shop receives a shipment of soccer balls 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 soccer balls and how many of basketballs does it need to buy to reach that amount?

We want to cover the floor of a recording studio that is a perfect square with rectangular soundproofing panels $18$ in wide by $24$ in long.

What must be the minimum side length of the room so that we can place 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 that measure $15$ in long, $10$ in wide and $6$ in high.

1. a) What must be the edge length 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?

Find a number between $300$ and $400$ that is, simultaneously, a multiple of $12$, $15$ and $18$. Explain the steps you have followed to find it using the concept of the least common multiple.

An action figure collector has fewer than $1.000$ pieces. He knows that if he organizes them into boxes of $18$, none are left over. If he organizes them into boxes of $20$, none are left over either, and if he groups them in groups of $24$ by $24$, the count is also exact.

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

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

1. a) First calculate the $m.c.m.(4, 6, 9)$.
2. b) Write the series of multiples of that value until you reach the indicated limit.
3. c) What is the largest multiple of these three numbers that is less than $500$?