To know the least common multiple of two numbers, they must be decomposed into prime factors.
Least common multiple (MCM) es un concepto que se utiliza en la math. El MCM entre varios natural numbers es el número natural más pequeño que es distinto de 0 y que resulta múltiplo de cada uno de ellos.
To calculate the LCM of two numbers , it is necessary to decompose them into prime factors. The LCM , therefore, will be the figure that we obtain from the multiplication of the non-common and common factors with elevation to the highest power.
Example of least common multiple
Let's see a practical example below to fully understand the procedure:
If we take the numbers 32 and 50, the first step will be to begin dividing each one by 2 until it is impossible to obtain an integer result , and then continue by 3, and so on until it can no longer be continued without entering the field. of real numbers . Starting with 32, we can divide it by 2, obtaining 16 and repeat this operation until we reach 1, having made 5 divisions, which tells us (in other words) that 32 is equal to raising 2 to its fifth power.
The remaining number is slightly more complicated, since we will have to change the divisor ; 50 divided by 2 gives us 25, which is not a multiple of 2 . Therefore, it will be necessary to look for a divisor that returns a quotient without a remainder , which in this case is the number 5. With it we can continue until we obtain the result 1, and by carefully observing the divisors, we can express 50 as the product of 2 times 5 squared. This is the time to compare the factors of both figures (32 and 50) and make a formula that includes all the factors resulting from both lists, raised to the highest power that we have obtained. In other words, the least common multiple of 32 and 50 is equal to the multiplication of 2 to the fifth power times 5 squared, which is 800.
Calculating the least common multiple is common when working with algebraic expressions.
Another way of calculation
In some cases, obtaining the MCM is very simple. The first step is to calculate the multiples of the numbers and then look for the first equivalence, going from smallest to largest (that is, the smallest number that is a multiple of the two and therefore appears in the two lists of multiples that we previously calculated).
If we want to discover the LCM of 3 and 5 , we will start by making a list of their multiples:
3 : 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33…
5 : 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55…
As can be seen, the first common multiple of 3 and 5 is 15 . Other common multiples of 3 and 5 are 30 , 45 , and 60 , for example.
Uses of the least common multiple
The LCM can be used to add fractions with unlike denominators. What we must do is consider the least common multiple of the denominators of the fractions and, after converting them into equivalent fractions, add them. In other words, suppose we must add the fractions 7/15 and 4/10; At first glance it can be seen that their denominators are different, which is why it is not possible to add their numerators. To solve this operation, as expressed above, it will first be necessary to make both fractions compatible.
With that objective, we must find the least common multiple of its denominators, which in this case is 30. Then, to convert its numerators, we will divide this value by each denominator and multiply its quotient by the numerator: (30 / 15) * 7 = 14 and (30 / 10) * 4 = 12 . Thus, with the fractions 14/30 and 12/30, all that remains is to add their numerators, which gives us the fraction 26/30 (note that the denominator remains intact).
Another use of the LCM is in the area of algebraic expressions . The LCM of two of these expressions is equivalent to the one with the smallest numerical coefficient and lowest degree that is susceptible to division by all the given expressions without a remainder.