Exercises: Lowest 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.

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 $LCM(12, 18)$.

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

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?

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).

Aoife and Cian train on the school running track. Aoife takes $6$ minutes to complete a full lap and Cian 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.

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

In Maths class, Aoife is trying to calculate the least common multiple of $12$ and $18$. To do so, she correctly finds their prime factorizations: $12 = 2^2 \times 3$ and $18 = 2 \times 3^2$. However, in the final step, she writes in her notebook:

"Since we are looking for the least, I must choose the common factors that have the lowest exponent. Therefore: $\text{l.c.m.}(12, 18) = 2 \times 3 = 6$".

1. a) Identify Aoife's flaw in reasoning and explain why the number $6$ could never be a common multiple of $12$ and $18$ based on the meaning of multiple.
2. b) Correct Aoife's method and calculate the correct result, explaining why the rule for the l.c.m. requires taking the factors raised to the highest exponent.

At the Santander bus station, two departures coincide at $07:00$ in the morning: the line to Torrelavega, which departs every $15$ minutes, and the line to El Astillero, which departs every $20$ minutes.

1. a) Before carrying out any calculation using prime factorization, make a reasoned estimate: is it possible for them to coincide again before $20$ minutes have passed? Could they take more than $15 \times 20 = 300$ minutes to depart together again? Explain between which logical bounds the coincidence time must lie.
2. b) Formally calculate the $\text{l.c.m.}(15, 20)$ to determine exactly how many minutes it will take for them to coincide again and at what time of the morning it will happen.
3. c) Check whether the result obtained matches your initial estimate and justify why the l.c.m. turns out to be considerably less than the product $15 \times 20$.

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) $m.c.m.(80, 120, 200)$

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 centre, 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 centre 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?

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 $3$.

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 $22:00$.

You have to find the least common multiple in two different cases:

• Case 1: $\text{l.c.m.}(14, 28)$

• Case 2: $\text{l.c.m.}(40, 60)$

You have three possible solving strategies available:

• Method A: Manually list the first few multiples of each number until finding the first one that matches.

• Method B: Find the prime factorization and take common and non-common factors to the highest exponent.

• Method C: Apply the direct divisibility relationship by observing whether one of the numbers is an exact multiple of the other.

1. a) For Case 1, indicate which of the three methods allows you to deduce the result most immediately and with almost no operations. Justify your choice and obtain the value.
2. b) For Case 2, compare the efficiency of Method A versus Method B: explain why Method A becomes impractical for numbers of a certain size and solve the case using the method you consider optimal.

At Connolly station (Dublin), three commuter trains start their service simultaneously at $06:00$ in the morning: Line 3 towards Maynooth has a departure frequency of every $15$ minutes, Line 4 towards Howth departs every $20$ minutes, and Line 5 towards Greystones departs every $12$ minutes.

1. a) Translate this situation into a mathematical model: justify what concept you need to calculate to find the moment when all three trains will depart together again and determine this time interval using prime factorization.
2. b) At what exact time in the morning will the next simultaneous departure of the three lines take place?
3. c) If the frequencies remain unchanged throughout the day and the service ends at $23:30$, calculate how many times in total the three simultaneous departures will coincide throughout the entire day (including the initial departure at $06:00$).

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 in exactly the same position again? At what time of day will this meeting take place?

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 $15$.

A design company wants to manufacture square billboard panels. To do this, they have three types of rectangular tiles of different colours, 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)?

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.)

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?

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?

Two unknown natural numbers, $A$ and $B$, simultaneously satisfy the following four conditions:

1. Their least common multiple is $\text{l.c.m.}(A, B) = 360$.

2. Neither number is divisible by the other.

3. Both numbers are strictly less than $360$.

4. Number $A$ is a multiple of $9$ and number $B$ is a multiple of $10$.

1. a) Decompose $360$ into prime factors and explain which powers of the prime factors ($2$, $3$, and $5$) must necessarily be present in $A$ or in $B$ so that their l.c.m. contains all the required ones.
2. b) Find two different pairs of numbers $(A, B)$ that satisfy all the given requirements and show through their prime factorizations that their l.c.m. is indeed $360$.
3. c) Could $A = 75$? Justify why it is impossible by analyzing only the prime factors of $75$ compared to those of $360$, without resorting to trial and error.

An agricultural cooperative in Navarra packs asparagus with protected designation of origin into glass jars. To prepare shipments, it uses boxes of three sizes: small boxes of $12$ jars, medium boxes of $18$ jars, and large boxes of $28$ jars. The cooperative orders standardized pallets on the condition that each pallet can be completely filled using exclusively $12$-jar boxes, exclusively $18$-jar boxes, or exclusively $28$-jar boxes, without mixing box models on the same pallet and without any jars left over.

1. a) Calculate the minimum number of jars that a pallet must hold to be compatible with all three box sizes.
2. b) The logistics manager proposes that the pallet have a capacity of $1500$ jars to make the most of the truck's volume. Prove, without performing division with decimals and using only the prime factorization of the l.c.m., why it is impossible for a pallet of $1500$ jars to satisfy the required condition.
3. c) If, due to weight and road safety regulations, the capacity of each pallet must strictly be between $2000$ and $3000$ jars, deduce with proper reasoning all possible capacities the pallet can have.