Example of Least Common Multiple

The Least Common Multiple, represented by the acronym m.c.m., of two or more numbers is the smallest of the common multiples of said numbers, other than zero. The easiest way to find the m.c.m. of two or more numbers is to decompose each of the numbers into its prime factors. So the least common multiple is equal to the product of all common and uncommon factors with their greatest exponent. We analyze the following example of the least common multiple in order to clarify the idea:
1) Let there be two ships that leave together from Mexico City. One will depart again within twelve (12) days and the other within forty (40) days. The question is, how many days will it take for both ships together to depart together?
In this example, what we must do is find the least common multiple of 12 and 40. To do this, we decompose each of these numbers into its prime factors.
No. Prime Factors
12 2
6 2
3 3
1
No. Prime Factors
40 2
20 2
10 2
5 5
1
In the example, decomposing a number into its prime factors represents dividing each of them by the smallest prime number that divides it exactly. So we come to the following conclusions:

12 = 2 x 2 x 3, or what is the same 12 = 2 squared (2) x3 y
40 = 2 x 2 x 2 x 5, or what is the same 40 = 2 cubed (3) x5
The Least Common Multiple is the product of the common and uncommon factors with their largest exponent, that is, the m.c.m. of 12 and 40 = 2 raised cubed x 3 x 5, m.c.m of 12 and 40 = 120, so the correct answer for this example is that the ships will come out together again within 120 days.

Another example of Least Common Multiple:

2) Two professional cyclists play a competition on the track of a velodrome. The first one takes 32 seconds to complete a complete lap and the second 48 seconds. How often in seconds will they meet at the starting point?
The example is similar to the previous one so we have to decompose 32 and 48 into their prime factors.
No. prime factors
32 2
16 2
8 2
4 2
2 2
1
No. prime factors
48 2
24 2
12 2
6 2
3 3
1
Therefore 32 = 2 x 2 x 2 x 2 x 2 that is 32 = 2 raised to the fifth (5) and 48 = 2 x 2 x 2 x 2 x 3 that is 48 = 2 raised to the fourth (4) x 3.
Since the least common multiple is equal to the producer of the common and uncommon factors with their greatest exponent, we have that the gcm of 32 and 48 = 2 raised to the fifth x 3. The least common multiple of 32 and 48 = 96, so the answer to this example is that the two cyclists will meet again at the starting point at 96 seconds.
3) In a banking house the security alarms are programmed efficiently. The first will sound every 10 seconds, the second every 15 seconds, and the last every 20 seconds. How many seconds will the alarms go off together?
The reasoning is similar to that of the previous examples, we must calculate the least common multiple of 10, 15 and 20. To do this, we perform the decomposition of its prime factors of the three numbers.
No. prime factors
10 2
5 5
1
No. prime factors
15 3
5 5
1
No. prime factors
20 2
10 2
5 5
1
We have that 10 = 2 x 5, that 15 = 3 x 5 and that 20 = 2 squared (2) x 5. The least common multiple of 10, 15, and 20 = 2 squared (2) x 3 x 5 = 60. The answer to this example is that all three alarms will sound together at 60 seconds (one minute).
Remember that prime numbers are those numbers that are only divisible between unity (1) and themselves.