Least Common Multiple

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.

A messaging app sends a "news summary" notification every $10$ minutes, while a social media app 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 the sets of multiples of $10$ and $15$ to find the lowest common multiple of both.

Without needing to perform prime factorization, reason what the $m.c.m.(5, 20)$ is. Take into account the divisibility relationship between both numbers. What happens to the lowest common multiple when one number is a multiple of another?

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) $m.c.m.(7, 11)$

Hint: Remember that if the numbers are coprime, their lowest common multiple is simply their product.

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

1. Perform the prime factorization of both numbers.

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

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

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

Find the least common multiple of the following groups of numbers using prime factorization:

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

Three smart lights in a shop 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 computer security system generates three automatic access codes. The first code changes every $2^3 \cdot 3 \cdot 5$ seconds, the second every $2^2 \cdot 3^2 \cdot 5$ seconds and the third every $2 \cdot 3 \cdot 5^2$ seconds.

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

Emma and Ethan are training on the school running track. Emma takes $6$ minutes to complete a full lap and Ethan 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 will have completed at that moment?

Two professional cyclists are training in a circular velodrome. The first completes a lap every $108$ seconds, while the second, who is slightly slower, takes $120$ seconds. If both pass the finish line together at this moment, how many minutes will it take for them to coincide at the same point for the first time?

The support team of a video game app 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 all three backups were made simultaneously today, in how many days will the same thing happen again?

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

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 again in exactly the same position? At what time of day will this encounter occur?

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 the three activities coincided today, how many times will they coincide on the same day over the next $365$ days?

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

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 the three sequencers flash at the same time, how many times will all three flash simultaneously again strictly before the session ends at $00:00$ hours (midnight)?

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

In the school library, they are reorganizing the fiction section. The librarian notices that if she stores the books in boxes of $6$, $8$ or $10$ copies, there are always exactly $3$ books left over. 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?

In a warehouse, we want to stack boxes of two different types of products. The boxes of product A have a height of $18$ in and those of product B have a height of $30$ in. What is the minimum height both stacks must reach so that they have exactly the same height? How many boxes of each type will there be at that point?

A design company wants to manufacture square advertising panels. To do this, they have three types of rectangular tiles of different colors, whose dimensions are $12 \times 18$ in, $15 \times 20$ in and $18 \times 24$ in. 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 in the same panel). What must be the side length of this panel? How many tiles of each type would be needed to cover it?

Three gears of 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 working and we mark a point of contact between the three, how many complete turns will each gear have made when they coincide for the first time in their initial position?

The product of two natural numbers is $6\,750$ and their greatest common factor ($M.C.D.$) is $15$. Using the relationship that exists between the product of two numbers, their $M.C.D.$ and their least 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 $15$.

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

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

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

(Hint: Factorize the known numbers into primes and use the definition of LCM and GCF.)