Least common multiple

To understand what the least common multiple is, it is important to refer to the word multiple . In the area of ​​mathematics, this term refers to all those natural numbers that can result from a multiplication of the number by other different numbers as long as they are natural.

Definition

The least common multiple is defined as the smallest number , except for the number 0 , which turns out to be a multiple of two or more numbers. It is important to remember that the multiple of a number are all those that you have to multiply by other different numbers, for example, those that we find in the multiplication tables. So, for example, we can say that the multiples of the number 2 are 3, 6, 9, 12, 15, 18, etc.

Of all these multiples that were mentioned above and following the definition of the least common multiple, we can define that 3 is the minimum of this number. Another important aspect that must be taken into account to understand the definition of the least common multiple , which occurs when we have two different numbers. If we obtain the numbers from these numbers through multiplication, it can be observed that some of them are repeated in the different numbers, these will then be the common multiples and the smallest number will be the least common multiple. It is represented by the letters mcm .

What is the least common multiple for?

The least common multiple or LCM is a concept used in the field of mathematics that serves to find the smallest natural number that results from the multiples of the numbers, always omitting the number 0. It can be used in the sum of fractions when these they have a denominator with different numbers. It also works in the field of algebra , specifically in algebraic expressions since in them, the least common multiple is equivalent to the numerical coefficient of smaller size and degree.

Properties

The properties that the least common multiple has are the following:

• When a corresponds to an integer, then [a, a] = a.
• If a and b are integers , [a, b] = b if, only if b turns out to be a multiple of a.
• (a, b) = [a, b] if they are equal or opposite.
• [a, b] = [ab] if, only if (a, b) = 1
• When the resulting product of two numbers is divided by their greatest common divisor, the quotient will then be the least common multiple.
• The least common multiple of two numbers in which the lesser divides the greater .
• When two numbers are relative primes , their LCM will be the product .
• When one number is a multiple of another , then the LCM will be the same number for both.
• When multiplying or dividing numbers, the LCM must also be multiplied or divided.
• All common multiples that are present in two or more numbers are multiples of the least common.

How to get 

In order to find the least common multiple of a number, two different mathematical operations can be performed:

• In the first way to find the least common multiple, it is a procedure in which all the multiples that each of the numbers that are being worked have are written , then the numbers that correspond to the common multiple are marked and finally, we choose the common multiple that represents the smallest number .
• The second way to find the least common multiple is done first, decomposing the numbers that are had into prime factors . Then we proceed to select those prime factors that are common and not common, taking into account which of them has the greater exponent. Then all the prime factors that are common must be selected and finally, the prime factors that have been selected are multiplied.

Example exercises and problems

Some examples of least common multiple are the following:

MCM of:

• (12, 8) = 24
• (6, 8) = 24
• (12, 18) = 36
• (3, 5) = 15