Exercises: Lowest common multiple

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.

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?

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?

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) $\text{m.c.m.}(20, 30)$

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?

To calculate the $\text{lcm}(6, 18)$, Lucía finds the prime factorisation of both numbers ($6 = 2 \cdot 3$ and $18 = 2 \cdot 3^2$) and applies the general rule of common and non-common factors with the highest exponent. In contrast, Marcos claims that there is no need to do any factorisation because it is obvious at a glance that the result is $18$.

1. a) Explain the theoretical property Marcos uses to give the answer immediately without needing to calculate.
2. b) Could Marcos use that same mental shortcut to directly calculate the $\text{lcm}(6, 15)$? Justify why in this second case it is indeed convenient to resort to Lucía's factorisation method.

Nerea has to calculate the $\text{lcm}(12, 15)$ to solve a class assignment. In her notebook she writes:

"Step 1: I find the prime factorisation: $12 = 2^2 \cdot 3$ and $15 = 3 \cdot 5$.

Step 2: I take the factors with the lowest exponent: the only factor that matches is $3$.

Result: $\text{lcm}(12, 15) = 3$".

1. a) Point out the two conceptual errors Nerea makes when applying the method. What mathematical concept has she actually calculated in her Step 2?
2. b) Correct the exercise in your notebook by finding the true result and explain why the number $3$ could never be a multiple of $12$ or of $15$.

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.

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.

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?

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?

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.

At an urban bus stop, two lines coincide at 08:00 in the morning. Circular line A arrives every $14$ minutes and intercity line B every $21$ minutes.

1. a) Before calculating the exact result, estimate and bound the interval: is it possible that they coincide again before $21$ minutes have passed? And after $294$ minutes ($14 \cdot 21$)? Explain why the $\text{lcm}$ of two numbers can never be less than the greater of them nor greater than their product.
2. b) Calculate the $\text{lcm}(14, 21)$ using prime factorisation and check if your estimation was correct.
3. c) At what exact time will they coincide at the stop again for the first time?

At the Méndez Álvaro commuter train station (Madrid), the departures of three trains coincide at 07:00 in the morning: a line C-1 train (departures every $12$ minutes), another from line C-5 (every $8$ minutes), and another from line C-10 (every $20$ minutes).

1. a) Model the situation by calculating how many minutes it will take for the three trains to depart simultaneously from that station again.
2. b) At what exact time in the morning will that next coincidence occur?
3. c) A notice on the station screens claims that the three trains will coincide four more times throughout that same morning, before 13:00. Mathematically check whether the information on the screens is true or false.

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?

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?

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 $500$?

The least common multiple of two natural numbers, $24$ and $b$, is equal to $120$. That is, $\text{lcm}(24, b) = 120$.

1. a) Find the prime factorisation of both the number $24$ and the result, $120$.
2. 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.
3. 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 $120$.
4. d) Could $b = 120$? Reason whether it is possible based on the divisibility relationship between $24$ and $120$.

A ceramics workshop in Talavera de la Reina manufactures rectangular tiles measuring $15\text{ cm}$ wide by $24\text{ cm}$ long. They want to place them on a wall, without cutting any piece and all oriented in the same direction, to form the smallest possible square surface.

1. a) Justify precisely why the side length of that square must be a common multiple of $15$ and $24$, and calculate the side length of the smallest square they can compose.
2. b) How many rectangular tiles will be needed in total to cover that first square? Justify your answer by relating the dimensions of the square to those of the tile.
3. c) A customer wants to order a larger square mural, but the workshop only has a box with $140$ tiles of that model. Reason whether they will have enough pieces to form the next possible square size without any tiles missing or left over from the mural.