Author:
Laura McKinney
Date Of Creation:
6 August 2021
Update Date:
1 July 2024
Content
Multiplicity is the product of a number with an integer. The least common multiple of a group of numbers is the smallest number that is divisible by all of them. To find the smallest common multiple, you need to determine the factor for each number. There are several different methods of finding the least common multiple, and they work for three or more numbers as well.
Steps
Method 1 of 4: Multiples enumeration
Review your numbers. This method is suitable for cases where two numbers that need to find a common multiple are both less than 10. For larger numbers, you should use another method.
- Take for example the problem of finding the smallest common multiple of 5 and 8. Since both numbers are small, it is very suitable for this method.
List the first few multiples of the first number. Multiplicity is the product of a number with an integer. In other words, they are the numbers that appear on your multiplication table.- For example, the first multiples of 5 are 5, 10, 15, 20, 25, 30, 35, and 40, respectively.
List the first few multiples of the second number. You should write near a list of multiples of the first for easy comparison.- For example, the first multiples of 8 include 8, 16, 24, 32, 40, 48, 56, and 64.
Find the least common multiple of the numbers above. You may have to add to the multiple list until you find a number that is both a multiple of one and a multiple of the other. That is your least common multiple.- For example, 40 is the smallest number that qualifies both as a multiple of 5 and a multiple of 8, so the minimum common multiple of 5 and 8 is 40.
Method 2 of 4: Analyze prime factors
Consider your numbers. This method is suitable for numbers greater than 10. For smaller numbers, you can use another method to find the smallest common multiple more quickly.
- For example, to find the minimum common multiple of 20 and 84, you should use this method.
Analysis of the first number. Here we will decompose this number into prime factors, that is, find prime numbers whose product is equal to the given number. To do this, a tree diagram can be used. After the analysis is complete, we will rewrite it in the form of an equation.
- For example, and, so the prime factors of 20 are 2, 2, and 5. Rewritten as an equation, we have:.
Analyze the second number. As with the first number, we find prime factors with the product of the second number.
- For example,,, and, so the prime factors of 84 are 2, 7, 3, and 2. Let's rewrite.
Write down the common factors. Establish multiplication of common factors. Cross off each factor common to the analytical equation to prime each time you remove it.
- For example, both numbers have a factor of 2, so we write and cross out a number 2 in both equations to be prime.
- Both numbers also share another factor of 2, so we will add and cross out the second factor 2 in each of the original analytical equations.
Add the remaining factors to the multiplication. Those are factors that are not crossed out after you have completed matching the two groups of factors. They are undivided factors.
- For example, in the equation, we have crossed out both 2s because they are also in the other number. And since there are 5 left, we will add the multiplication:.
- In the equation, we have also crossed out both 2. There are 7 and 3 left, so we will add the multiplication:.
Minimum common multiple. To do this we simply multiply the numbers in the multiplication we just created.
- For example: . So the minimum common multiple of 20 and 84 is 420.
Method 3 of 4: Use a grid or ladder method
Draw a checkered grid. Caro grid consists of two sets of parallel lines perpendicular to each other. They form three columns and look like a pound sign (#) on a phone or keyboard. Write the first number in the top, center box. Write the second number in the top right box.
- For example, with the problem of finding the minimum common multiple of 18 and 30, we write 18 at the top, the center of the grid to 30 in the upper right.
Find some common factor of both numbers. Write this number in the top left box. It's not required, but it's better if the factor is prime.
- In the example problem, since 18 and 30 are even, 2 is their common factor. Therefore, we will write 2 in the upper left cell of the grid.
Divide each number by the factor you just found and write the quotient in the box below. Loving is the result of division.
- So 9 would be written under 18.
- , so 15 should be written under 30.
Find the common factor of two traders. If there are no more common factors, you can skip it and go to the next step. If there is a common factor, we will write it in the left middle cell of the grid.
- For example, 9 and 15 are both divisible by 3, so we will write 3 in the left middle cell of the grid.
Divide the quotient by this common factor. Write a new spear under the first spear.
- so 3 should be written under 9.
- so 5 should be written under 15.
Expand the mesh if necessary. Continue like that until the two spears have no common factors.
Circle the numbers on the first and last row of the grid, forming an “L”. Set the whole multiplication of these factors.
- For example because 2 and 3 are in the first column and 3 and 5 are in the last row, we have.
Complete multiplication. By multiplying these numbers, we obtain the minimum common multiple of the two given numbers.
- Eg . Therefore, 90 is the minimum common multiple of 18 and 30.
Method 4 of 4: Using Euclidean algorithm
Understand the terminology used in division. The divisor is the number given to divide. Divisor is the number by which the divisor is divided. Loving is the answer of division. Balance is what is left over after division.
- For example, in the residual equation:
15 is the dividend
6 is the divisor
2 is spear
3 is the balance.
- For example, in the residual equation:
Set up the quotient-residue formula. These are: dividend = divisor x quotient + remainder. You will use it to set up the Euclidean algorithm to find the greatest common divisor of two given numbers.
- Eg .
- The greatest common divisor is the divisor, or the greatest factor, of both numbers.
- In this method, we will first find the greatest common divisor and then use it to find the smallest common multiple.
The larger number is the divisor, the smaller the divisor. Set up the quotient-balance equation for these two numbers.
- For example, with the problem of finding the least common multiple of 210 and 45, we will calculate.
Take the original divisor as the new divisor, and the original balance as the new divisor. Set up the quotient-balance equation for these two numbers.
- For example: .
Repeat until the balance is 0. For each new equation, use the divisor of the previous equation as the divisor and the previous remainder as the divisor.
- For example: . Since the balance is zero, we will stop here.
Look at the final divisor. This is the greatest common divisor of the initial two numbers.
- In the example problem, since the last equation is and the final divisor is 15, 15 is the greatest common divisor of 210 and 45.
Multiply two numbers. Divide the product by their greatest common divisor. The result is the minimum common multiple of the two given numbers.
- For example: . Divide by the greatest common divisor, we get:. So 630 is the minimum common multiple of 210 and 45.
Advice
- To find the smallest common multiple of three or more numbers, you can adjust the above methods a bit. For example, to find the least common multiple of 16, 20 and 32, you can find the lowest common multiple of 16 and 20 first (which is 80), and then find the minimum common multiple of 80 and 32 to get the result. and finally 160.
- The smallest common multiple is frequently used. The most common is in fraction addition and subtraction: fractions must have the same denominator and therefore, if they are different from the sample, you will have to converge the denominator to perform the calculation. The best way is to find the lowest common denominator - the least common multiple of the denominators.
