Easy Way to Find the Lcm of Two Numbers

August 06, 2022 Post a Comment

Download Article

A multiple is the result of multiplying a number by an integer. The least common multiple (LCM) of a group of numbers is the smallest number that is a multiple of all the numbers. To find the least common multiple you need to be able to identify the factors of the numbers you are working with. You can use a few different methods to find the least common multiple. These methods also work when finding the LCM of more than two numbers.

1
Assess your numbers. This method works best when you are working with two numbers that are less than 10. If you are working with larger numbers, it's best to use a different method.
- For example, you might need to find the least common multiple of 5 and 8. Since these are small numbers, it is appropriate to use this method.
2
Write out the first several multiples of the first number. A multiple is a product of any number and an integer.^[1] In other words, they are the numbers you would see in a multiplication table.
- For example, the first several multiples of 5 are 5, 10, 15, 20, 25, 30, 35, and 40.
Advertisement
3
Write out the first several multiples of the second number. Do this near the first set of multiples, so that they are easy to compare.
- For example, the first several multiples of 8 are 8, 16, 24, 32, 40, 48, 56, and 64.
4
Find the smallest multiple the numbers have in common. You might need to extend your list of multiples until you find one both numbers share. This number will be your least common multiple.^[2]
- For example, the lowest multiple 5 and 8 share is 40, so the least common multiple of 5 and 8 is 40.

1
Assess your numbers. This method works best when both of the numbers you are working with are greater than 10. If you have smaller numbers, you can use a different method to find the least common multiple more quickly.
- For example, if you need to find the least common multiple of 20 and 84, you should use this method.
2

Factor the first number. You want to factor the number into its prime factors; that is, find the prime factors you can multiply together to get this number. One way to do this is by creating a factor tree. Once you are done factoring, rewrite the prime factors as an equation.
3

Factor the second number. Do this in the same way you factored the first number, finding the prime factors you can multiply together to get the number.
4

Write down the factors each number shares. Write the factors as a multiplication sentence. As you write each factor, cross it off in each numbers factorization equation.
5

Add any leftover factors to the multiplication sentence. These are the factors you did not cross out when comparing the two groups of factors. Thus, these are factors that the two numbers do not share.^[3]
6
Calculate the least common multiple. To do this, multiply together all of the factors in your multiplication sentence.
- For example, $2\times 2\times 5\times 7\times 3=420$ . So, the least common multiple of 20 and 84 is 420.

1
Draw a tic-tac-toe grid. A tic-tac-toe grid is two sets of parallel lines that intersect each other perpendicularly. The lines form three rows and three columns and looks like the pound key (#) on a phone or keyboard. Write your first number in the top-center square of the grid. Write your second number in the top-right square of the grid.^[4]
- For example, if you are trying to find the least common multiple of 18 and 30, write 18 in the top center of your grid, and 30 in the top right of your grid.
2
Look for a factor that is common to both numbers. Write this number in the top-left square of your grid. It is helpful to use prime factors, but you don't necessarily have to.
- For example, since 18 and 30 are both even numbers, you know that that they both have a factor of 2. So write 2 in the top-left of the grid.
3

Divide the factor into each number. Write the quotient in the square below either number. A quotient is the answer to a division problem.
4
Find a factor that is common to the two quotients. If there is no factor common to both quotients, you can skip this and the next step. If there is a common factor, write it in the middle-left square of the grid.
- For example, 9 and 15 both have a factor of 3, so you would write 3 in the middle-left of the grid.
5

Divide this new factor into each quotient. Write this new quotient below the first ones.
6

Extend your grid if necessary. Follow this same process until you reach a point where the last set of quotients have no common factor.
7
Draw a circle around the numbers in the first column and last row of your grid. You can think of it as drawing an "L" for "least common multiple." Write a multiplication sentence using all of these factors.^[5]
- For example, since 2 and 3 are in the first column of the grid, and 3 and 5 are in the last row of the grid, you would write the sentence $2\times 3\times 3\times 5$ .
8
Complete the multiplication. When you multiply all of these factors together, the result is the least common multiple of your two original numbers.^[6]
- For example, $2\times 3\times 3\times 5=90$ . So, the least common multiple of 18 and 30 is 90.

1
Understand the vocabulary of division. The dividend is the number being divided. The divisor is the number the dividend is being divided by. The quotient is the answer to the division problem. The remainder is the amount left over after a number is divided by another.^[7]
- For example, in the equation $15\div 6=2\;{\text{remainder}}\;3$ :
  15 is the dividend
  6 is the divisor
  2 is the quotient
  3 is the remainder.
2
Set up the formula for the quotient-remainder form. The formula is ${\text{dividend}}={\text{divisor}}\times {\text{quotient}}+{\text{remainder}}$ .^[8] You will use this form to set up Euclid's algorithm to find the greatest common divisor of two numbers.
- For example, $15=6\times 2+3$ .
- The greatest common divisor is the largest divisor, or factor, that two numbers share.^[9]
- In this method, you first find the greatest common divisor, and then use it to find the least common multiple.
3
Use the larger of the two numbers as the dividend. Use the smaller of the two numbers as the divisor. Set up an equation in quotient-remainder form for these two numbers.
- For example, if you are trying to find the least common multiple of 210 and 45, you would calculate $210=45\times 4+30$ .
4
Use the original divisor as the new dividend. Use the remainder as the new divisor. Set up an equation in quotient-remainder form for these two numbers.
- For example, $45=30\times 1+15$ .
5
Repeat this process until you have a remainder of 0. For each new equation, use the previous equation's divisor as the new dividend, and the previous remainder as the new divisor.^[10]
- For example, $30=15\times 2+0$ . Since the remainder is 0, you do not need to divide any further.
6
Look at the last divisor you used. This is the greatest common divisor for the two numbers.^[11]
- For example, since the last equation was $30=15\times 2+0$ , the last divisor was 15, and so 15 is the greatest common divisor of 210 and 45.
7

Multiply the two numbers. Divide the product by the greatest common divisor. This will give you the least common multiple of the two numbers.^[12]

Add New Question

Question

What's the formula of the least common multiple?

This answer was written by one of our trained team of researchers who validated it for accuracy and comprehensiveness.

wikiHow Staff Editor

Staff Answer

The formula is lcm(a, b) = a × b / gcd(a, b), where a and b are the numbers for which you want to find the LCM, and GCD is the greatest common divisor.
Question

Is there a least common multiple calculator?

This answer was written by one of our trained team of researchers who validated it for accuracy and comprehensiveness.

wikiHow Staff Editor

Staff Answer

Yes, there are multiple LCM calculators online. Try websites like CalculatorSoup.com or Calculator.net to find calculators for finding the LCM and doing a variety of other common calculations.
Question

What's the fastest way to find the least common multiple of two numbers?

This answer was written by one of our trained team of researchers who validated it for accuracy and comprehensiveness.

wikiHow Staff Editor

Staff Answer

One quick and easy way to do it is to start by finding the greatest common factor (GCF) of the 2 numbers. Divide the GCF into either one of the 2 numbers, then multiply the result by the other number. This will give you the LCM.
Question

What is the least common multiple of 105, 210 and 630?

The least common multiple of 105, 210 and 630 is 630. Both 105 and 210 are factors of 630, therefore 630 is the least common multiple of the three numbers. To verify this, see that 105 x 6 = 630, 210 x 3 = 630, and 630 x 1 = 630.
Question

How do I calculate the least common multiple and lowest common denominator of fractional numbers?

To find the lowest common denominator of a fraction, you need to find the LCM of the two denominators.
Question

What are the two numbers that have the least common multiple of 20?

Since 20 x 1 = 20, 20 is a multiple of 20. That means 10 and 20 are the two numbers that have an LCM of 20.
Question

What is the lowest common multiple of 16 and 20?

16 = 2 x 2 x 2 x 2 20 = 2 x 2 x 5 Since both share a factor of 2 twice, begin by writing 2 x 2. 16 two factors of 2 left over, so the multiplication sentence becomes 2 x 2 x 2 x 2 20 has a factor of 5 left over, so the multiplication sentence becomes 2 x 2 x 2 x 2 x 5 = 80. So, the LCM of 16 and 20 is 80.
Question

I applied Method 1 to the numbers from Method 4: 210 and 45. I came up with an LCD of 420, which is not right. Where did I go wrong? (210=2x3x5x7, 45=3x3x5) leading to (1x2, 2x3, 1x5, 1x7) leading to 420.

Orangejews

Community Answer

You are right that the LCM is 2×3×3×5×7. But that comes out to 630, not 420. Perhaps you accidentally wrote a 2 instead of 3 at some point while multiplying out that last step.
Question

What is the LCM for (a-n) and (a-n)²?

(a-n)².
Question

Why in the first method is 42 used instead of 2?

Using 2 instead of 42 in the first method would be too easy to solve. Since 2 is a factor of 20, the first multiple that is common to both of them will be 20. This is because if a number n is a factor of a number p, then the LCM of n and p will always be P. Using the number 42 really shows that we commonly have to solve a problem in which two numbers are not relatively prime, but is relatively large as to pose a challenge to us.

See more answers

Ask a Question

200 characters left

Include your email address to get a message when this question is answered.

Submit

If you need to find the LCM of more than two numbers, the above methods can be tweaked. For instance, to find the LCM of 16, 20, and 32, you could start by finding the LCM of 16 and 20 (which is 80), and then find the LCM of 80 and 32, which turns out to be 160.
The LCM has many uses. The most common is that, whenever you add or subtract fractions, they must have the same denominator; if they do not, you need to convert each fraction to some equivalent fraction so they will share the same denominator. The best way to do that is to find the lowest common denominator (LCD) -- which is just the LCM of the denominators.

Thanks for submitting a tip for review!

About This Article

Article Summary X

To find the least common multiples of two numbers, start by writing out the first several multiples for each number. For example, the first several multiples of 5 would be 5, 10, 15, 20, 25, 30, 35, and 40. Once you've written out the first several multiples for both numbers, find the smallest multiple that they have in common, which is the least common multiple. If they don't have a common multiple, keep listing the multiples for each number until you find one. If you want to know how to use prime factorization or an algorithm to find the least common multiple, keep reading the article!

Did this summary help you?

Thanks to all authors for creating a page that has been read 1,209,627 times.

Did this article help you?

carterwhoduch.blogspot.com

Source: https://www.wikihow.com/Find-the-Least-Common-Multiple-of-Two-Numbers

Carter Whoduch

Easy Way to Find the Lcm of Two Numbers

About This Article

Did this article help you?

Post a Comment for "Easy Way to Find the Lcm of Two Numbers"