Least Common Multiple

Find the lowest common multiple of the following pairs of numbers mentally:

1. a) $m.c.m.(2, 5)$
2. b) $m.c.m.(3, 9)$
3. c) $m.c.m.(4, 10)$
4. d) $m.c.m.(7, 11)$

Without needing to find the prime factorisation, deduce what the $m.c.m.(5, 20)$ is. Consider the divisibility relationship between both numbers. What happens to the lowest common multiple when one number is a multiple of another?

In a platform video game, your character gets a protective shield every $4$ levels and a speed boost every $6$ levels. If you have just started the game at level $0$, at which level will both rewards coincide for the first time? Calculate the result by finding the first multiples of each number.

The number line shows the first jump of each reward. Continue the jumps yourself until you find the first level where they coincide.

A messaging application sends a "news digest" notification every $10$ minutes, while another social media application does so every $15$ minutes. If both sent a notification at $10:00$ in the morning, how many minutes will pass until they coincide again for the first time? Write out the sets of multiples of $10$ and $15$ to find the lowest common multiple.

Calculate the lowest common multiple of the numbers $12$ and $18$ in your exercise book using prime factorisation. To do this, follow these steps:

1. Find the prime factorisation of both numbers.

2. Identify the common and non-common factors raised to the highest power.

3. Multiply them to obtain the $m.c.m.(12, 18)$.

Find the lowest common multiple of the following groups of numbers using prime factorisation:

1. a) $m.c.m.(45, 75)$
2. b) $m.c.m.(12, 18, 24)$
3. c) $m.c.m.(80, 120, 200)$

Find the smallest number that, when divided by $15$, $20$ and $35$, always leaves a remainder equal to $0$. Once found, state what the smallest number would be that, when divided by those same values, would always leave a remainder of $3$.

Find the value of a natural number $x$, knowing that it is less than $50$ and that these two conditions are simultaneously met:

• The lowest common multiple of $24$ and $x$ is $120$.

• The highest common factor of $24$ and $x$ is $4$.

(Hint: Find the prime factorisations of the known numbers and use the definition of LCM and HCF.)

A computer security system generates three automatic access codes. The first code changes every $2^3 \times 3 \times 5$ seconds, the second every $2^2 \times 3^2 \times 5$ seconds and the third every $2 \times 3 \times 5^2$ seconds.

1. a) Express the time between changes for each code as a decimal.
2. b) Calculate, expressing the result using its prime factorisation, how often all three code changes coincide simultaneously.
3. c) How many minutes will have passed between each coincidence?

The product of two natural numbers is $6\,750$ and their highest common factor ($M.C.D.$) is $15$. Using the relationship between the product of two numbers, their $M.C.D.$ and their lowest common multiple ($m.c.m.$), first calculate the value of the $m.c.m.$ of these numbers. Next, find out what those two numbers are, knowing that their difference is exactly $15$.

Three smart lights in a store window are programmed to change color at different intervals. The first changes every $2$ seconds, the second every $3$ seconds and the third every $4$ seconds. If all three are turned on at the same time, how long will it take for them to change color simultaneously again?

A DJ uses two drum loops of different lengths. The first loop lasts $8$ bars and the second $12$ bars. How many bars will pass before both loops restart at the same time? Use whichever method you prefer (listing multiples or prime factorisation).

Lucía and Marcos train on the school running track. Lucía takes $6$ minutes to complete a full lap and Marcos takes $8$ minutes. If they start together from the finish line:

1. a) After how many minutes will they cross the finish line at the same time again?
2. b) Calculate how many laps each person will have completed at that moment.

Two professional cyclists train on a circular velodrome. The first completes a lap every $108$ seconds, while the second, who goes a bit slower, takes $120$ seconds. If both pass the finish line together right now, how many minutes will it take for them to coincide at the same point again for the first time?

In a logistics center, one maintenance alarm sounds every $45$ minutes and another every $60$ minutes. If at $08:00$ in the morning both alarms coincided, at what time will they sound together for the second time after $08:00$?

A cultural center organizes three different activities: robotics classes every $12$ days, drama workshops every $15$ days and film club sessions every $20$ days. If all three activities coincided today, how many times will they coincide on the same day over the next $365$ days?

In a warehouse we want to stack boxes of two different products. The boxes of product A are $18$ cm tall and the boxes of product B are $30$ cm tall. In the figure you can see one box of each type (the heights are drawn to the same scale).

If we stack boxes of product A in one pile and boxes of product B in another:

1. a) What is the minimum height both piles must reach so that they have exactly the same height?

1. b) How many boxes of each type will there be in each pile at that point?

The support team for a video game application has three servers that perform automatic backups. The first server makes a backup every $24$ hours, the second every $36$ hours and the third every $48$ hours. If today all three backups were performed simultaneously, in how many days will the same happen again?

A transport company has two intercity routes. The North route bus passes by the central station every $1$ hour and $15$ minutes. The South route bus passes every $50$ minutes. If at $10:00$ in the morning both departed at the same time, calculate how many times they will have coincided at the station before the day ends at $22:00$.

A design company wants to manufacture square billboard panels. To do this, they have three types of rectangular tiles of different colors, whose dimensions are $12 \times 18$ cm, $15 \times 20$ cm and $18 \times 24$ cm. They want to build the smallest possible square panel that can be covered by any of the three types of tiles independently (without mixing them on the same panel). What should the side length of this panel be? How many tiles of each type would be needed to cover it?

In the school library, they are reorganising the fiction section. The librarian notices that if she puts the books in boxes of $6$, $8$ or $10$ copies, there are always exactly $3$ books left over outside. Knowing that the total number of books in the section is between $400$ and $600$, how many books are there exactly in the fiction section?

A DJ has three light sequencers programmed to flash at different intervals. The first flashes every $24$ seconds, the second every $36$ seconds and the third every $40$ seconds. If at $23:00$ hours all three sequencers flash together, how many times will all three coincide simultaneously strictly before the set ends at $00:00$ hours (midnight)?

Three gears in an industrial machine are connected. Gear $A$ has $45$ teeth, gear $B$ has $60$ teeth and gear $C$ has $75$ teeth. If the machine starts operating and we mark a point of contact between all three, how many full turns will each gear have completed when they coincide in their initial position for the first time?

In a space station, four research satellites orbit an asteroid. The first completes its orbit every $12$ hours, the second every $18$ hours, the third every $20$ hours and the fourth every $30$ hours. If today at $12:00$ noon they are aligned for the first time, how many full days must pass for them to coincide in exactly the same position again? At what time of day will this meeting take place?