O minimum common multiple (MMC) between two integers x and y is the smallest integer that is a multiple of x and y simultaneously. In this way, there is at least one way to find the MMC between two numbers x and y: search the sets of multiples of x and y for the smallest common element. Of course, there is a practical method for finding this number, which will be discussed below. However, it is necessary to understand the concept of multiples of an integer well.
What are multiples?
An integer k is called a multiple of x if there is some natural number n such that n·x = k. Take the example of the number 110. He is multiple of 10, since 110 is the result of multiplying 10 by the natural number 11.
In this way, it is possible to identify whether the integer k is multiple of x by trial and error or by doing the inverse operation of multiplication (division). The number k is a multiple of x if there is a natural number n such that:
n = k
x
In other words, to find out if 110 is a multiple of 10, divide 110 by 10. If the result found is a natural number, 110 is a multiple of 10; otherwise, no.
As the set of natural numbers is infinite, the set of multiples of any integer is also infinite. However, to solve exercises involving multiple and MMC, it is good to write a list of the first multiples of a number to get a better analysis of the behavior of its multiples.
Below is a list of the first 10 multiples of 8, 10, 12, 20 and 40. They are the first 10 because they are the result of multiplying these numbers with the first 10 natural numbers.
10 first naturals: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80
Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
Multiples of 20: 20, 40, 60, 80, 100, 120, 140, 160, 180, 200
Multiples of 40: 40, 80, 120, 160, 200, 240, 280, 320, 360, 400
Least common multiple
To find the least common multiple between two numbers, find the minor multiple that they have in common. The first technique used to find the mmc is to look for it between multiples of the two numbers. Look at the example:
The least common multiple between 10 and 12 is 60, because between the multiples of 10 and 12, 60 is the smallest number that is a multiple of both. Watch:
Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
For these two numbers, which are small, it is easy to find the MMC. But what about when the calculation of MMC between 256 and 384 is required? Numerous tiring multiplications will be needed if you want to proceed by this method. For that, there is a practical method which will be discussed below.
Decomposition method for calculating MMC
To calculate the least common multiple between two numbers, you can make the prime factor decomposition their. For example, the decompositions into prime factors of 10 and 12 are:
10 = 2·5
12 = 2·2·3 = 22·3
Note: Whenever repeated factors appear, write them in power form, as was done in the decomposition of number 12.
The MMC between 10 and 12 will be the product of the prime factors, except for the repeating factors that have the smallest exponent. Thus, the minimum will be:
22·3·5 = 4·3·5 = 12·5 = 60
Note that factor 2, from the decomposition of number 10, was ignored, as the same factor, from decomposition of number 12, was squared.
This makes calculating the MMC between 256 and 384 easier. Look:
256 = 2·2·2·2·2·2·2·2 = 28
384 = 2·2·2·2·2·2·2·3 = 27·3
MMC will be product 28·3 = 256·3 = 768.
Example 2: MMC between 768 and 4608
768 = 28·3
4608 = 29·32
The MMC will be the product: 29·32.
Example 3: Calculate the MMC between 2700 and 4608
2700 = 33·22·52
4608 = 29·32
Note that the factors are 2, 3 and 5. Those with the highest exponents are 29, 33 and 52. So the MMC will be:
29·33·52 = 345600
Practical method to calculate MMC
It is possible to note that to decompose numbers into prime factors, it is necessary to divide them by the smallest possible prime divisor and still ignore the factors that are repeated in the same division. There is a method capable of doing this task. To teach you, we will use the example of MMC between 1000 and 1024.
Write these two numbers side by side, separated by a comma, and pass a vertical side stroke to the right of them:
1000, 1024 |
|
|
To the right of this trace, write the smallest prime number that divides at least one between 1000 and 1024. In this case, the number is 2 and it divides both.
1000, 1024 | 2
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|
Just below each of them, write the result of your division by 2 and, for these results, repeat the procedure above until it is no longer possible to divide either number by 2.
1000, 1024 |2
500, 512 |2
250, 256 |2
125, 128 |2
125, 64|2
125, 32 |2
125, 16 |2
125, 8 |2
125, 4 |2
125, 2 |2
125, 1 |
Note that at one point we find the result 125 in the 1000 column, but 125 is not divisible by 2. In the column number 1024, we only get results divisible by 2. In this case, we continue to divide the numbers in the 1024 column by 2 and repeat the number 125.
When the numbers in both the 1000 and 1024 columns are no longer divisible by 2, try the next prime: the number 3. When there are no more divisors of 3, try the next one and so on until you get the result “1,1”. In the case of the example, 125 is not divisible by 3, but by 5, so we'll repeat the process by putting 5 to the right of the dash. Watch:
1000, 1024 |2
500, 512 |2
250, 256 |2
125, 128 |2
125, 64|2
125, 32 |2
125, 16 |2
125, 8 |2
125, 4 |2
125, 2 |2
125, 1 |5
25, 1 |5
5, 1 |5
1, 1 |
Once that's done, multiply the factors found to the right of the vertical line:
2·2·2·2·2·2·2·2·2·2·5·5·5 = 210·53 = 128000
Example 2: Calculate the MMC between 432 and 384:
432, 384 |2
216, 192 |2
108, 96 |2
54, 48 |2
27, 24 |2
27, 12 |2
27, 6 |2
27, 3 |3
9, 1 |3
3, 1 |3
1, 1 |
The MMC will be: =
2·2·2·2·2·2·2·3·3·3 = 27·33 = 128·9 = 1152
To calculate the MMC of three numbers or more, simply use the practical method discussed here, putting all these numbers side by side.
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/matematica/minimo-multiplo-comum-mmc.htm
