Lowest common multiple
One alarm sounds every $12$ minutes, another every $15$ and another every $20$. All three have just gone off together. When will they next sound together? That is exactly the question that the lowest common multiple answers.
What the l.c.m. is
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…
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…
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.
Calculating the l.c.m. by prime factor decomposition
Listing multiples is slow for large numbers. The quick method: break each number down into prime factors and take all the factors, common and non-common, raised to the highest power.
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…
Calculate the lowest common multiple of the numbers $12$ and $18$ in your exercise book using prime factorisation. To do this, follow these steps:
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Find the prime factorisation of both numbers.
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Identify the common and non-common factors raised to the highest power.
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Multiply them to obtain the $m.c.m.(12, 18)$.
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…
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 $08:00$?
The l.c.m. of three or more numbers
The method is the same, except that you compare more factorisations at once.
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…
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?
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…
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 $15$.
Applications of the l.c.m.
The l.c.m. appears whenever something repeats with a period and you ask when the repetitions coincide.
Find the lowest common multiple of the following groups of numbers using prime factorisation: a) $m.c.m.(45, 75)$ b) $m.c.m.(12, 18, 24)$ c) $m.c.m.(80, 120, 200)$
Find the lowest common multiple of the following groups of numbers using prime factorisation:
- a) $m.c.m.(45, 75)$
- b) $m.c.m.(12, 18, 24)$
- c) $m.c.m.(80, 120, 200)$
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…
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?
Back to the start: Since l.c.m.$(12, 15, 20) = 60$, the three alarms will sound together again in $60$ minutes (exactly one hour later).