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.

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

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 $15 \times 20$.

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 $LCM(12, 18)$.

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.

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 $3$.

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?

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?

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 $15$.

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.

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 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?