O least common multiple (MMC) between whole numbers is the smallest number, also an integer, which is multiple of all these numbers at the same time. For example, the MMC between 2 and 12 is 12, because the multiples of 2 are 2, 4, 6, 8, 10, 12… and the multiples of 12 are: 12, 24, …
In other words, consider a set A of natural numbers non-negative and sets A1, A2, … formed by the multiples of each of the elements of set A. The smallest common element within sets A1, A2, … it's the Minimummultiplecommon of the elements of set A. In other words, the smallest element of intersection A1 ∩ A2 ∩ A2 ∩… is A's MMC.
This definition and the example given before it illustrate one of the methods that can be used to find the MMC of a set of numbers.
The notation used to represent the Minimummultiplecommon is: MMC(a, b, c) = d, where “d” is the MMC of “a”, “b” and “c”.
See too: What are numeric sets?
Finding the Least Common Multiple
The most basic method that can be used to find the Minimummultiplecommon between two or more numbers is to write yours
multiples until you find the first one that is common to all the observed numbers.O MMC between numbers 2, 4 and 12 can be found by doing:
M(2) = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, …}
M(4) = {4, 8, 12, 16, 20, 24, …}
M(12) = {12, 24, 36, 48, …}
Note that the intersection between the three sets of multiples is:
M(2) ∩ M(4) ∩ M(12) = {12, 24, …}
The smallest number of this intersection is 12, so MMC(2, 4, 12) = 12.
We can also simplify the thinking and just point the number 12 as “smallermultiple 2, 4 and 12”, avoiding the need to include the intersection between sets of multiples in the solution.
Practical method for calculating the least common multiple
O methodpractical to calculate the least common multiple is based on the factor decompositioncousins these numbers, but there is an algorithm that can make it easier to find it.
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This algorithm it consists of placing the numbers whose MMC will be calculated side by side and separated by a comma. Then we find the smallest prime number that divides at least one of them and we perform the division, placing the result just below it. If any of the elements is not divisible by this number, just repeat it in place of the result. This process is repeated until the result of all divisions is 1. O MMC it will be the product of all the prime numbers used in the divisions.
See an example:
To find the Minimummultiplecommon between 144, 26 and 10, we will do:
144, 26, 10 | 2
72, 13, 5 | 2
36, 13, 5 | 2
18, 13, 5 | 2
9, 13, 5 | 3
3, 13, 5 | 3
1, 13, 5 | 5
1, 13, 1 | 13
1, 1, 1 |
Hence, MMC(144, 26, 10) = 2·2·2·2·3·3·5·13 = 9360.
MMC characteristics and properties
The following list shows some features of the Minimummultiplecommon and then some of the properties of this operation.
1 - The MMC could also be written in factored form 24·32·5·13.
2 – When doing the decompositioninfactorscousins of the three numbers, we will find:
144 = 24·32
26 = 2·13
10 = 2·5
So the Minimummultiplecommon it can be defined as the product of the prime factors of the numbers excluding those that have the smallest exponent.
Note, for example, that both 144, 26 and 10 have a prime factor of 2, but only 2 was used in MMC4, which is the one that has the greatest exponent.
3 – The previous observation leads to the following ones properties:
The) MMC(a, a, … a) = a
B) MMC(the, the2, a3, …, Theno) = theno
ç) MMC between numbers that are prime to each other, that is, that do not have prime factors in common, is always equal to 1.
of MMC between numbers that are multiple is always the largest among them. The MMC of 5 and 10, for example, is 10.
By Luis Paulo Silva
Graduated in Mathematics
