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wikiHow to Find the Least Common Multiple of Two Numbers

Four Methods:Use the Prime FactorizationList all the Multiples of Both NumbersUse a Common Factors GridUse Euclid's AlgorithmCommunity Q&A

The Least Common Multiple (LCM) of a group of numbers is the smallest number that is a multiple of all the numbers. For instance, the LCM of 16 and 20 is 80; 80 is the smallest number that is both a multiple of 16 and a multiple of 20. You can find the LCM of two or more numbers through various methods. If you would like to know how to find the LCM of two or more numbers, just follow these steps.

Method 1
Use the Prime Factorization

  1. 1
    Find the prime factorization of both numbers. This is an ideal method for larger numbers. The first step to finding the least common multiple of two numbers using this method is factoring both numbers down to the prime numbers that are multiplied to create that number as a product. You can start by just listing two numbers (factors) that multiply to create that number and then factoring them down to their prime components.[1] Let's say you're looking for the least common multiple of 20 and 42. Here's how you would factor them 20 = 2 x 2 x 5 42 = 2 x 3 x 7
  2. 2
    Write down the most times any prime number occurs in the prime factorization of each number. If the number just occurs in one number, then it has one occurrence. Here is a list of the most occurrences of each prime number from the previous example 2 → 2 times 3 → 1 time 5 → 1 time 7 → 1 time
  3. 3
    Multiply all the prime factors together -- multiplying each number once for each of its occurrences. Since 2 occurs twice, you'll have to multiply it twice. Here's what you should do to find the LCM:

Method 2
List all the Multiples of Both Numbers

  1. 1
    List a few multiples of the first number in ascending order. This is an ideal method for smaller numbers, especially numbers lower than 10. It is not recommended for larger numbers since it can get sticky. Let's say you're looking for the LCM of 5 and 8. First, list a few of the multiples of 5 5 x 1 = 5 5 x 2 = 10 5 x 3 = 15
  2. 2
    List a few multiples of the second number in ascending order. Now do the same thing for the number 8.
    • 8 x 1 = 8

      8 x 2 = 16

      8 x 3 = 24
  3. 3
    Alternate between listing a few multiples of both numbers until you have found the lowest common multiple. In some cases, you will find the LCM after listing just a few of the multiples of each number. In this case, keep going until you reach the same multiple with both 5 and 8. That number will be your LCM

    5 x 4 = 20

    5 x 5 = 25

    5 x 6 = 30

    5 x 7 = 35

    5 x 8 = 40

    8 x 4 = 32

    8 x 5 = 40

Method 3
Use a Common Factors Grid

  1. 1
    Write the numbers at the top of the Common Factors Grid. Leave a small space to the left of the numbers and as much space as you can below the numbers. Let's say we're working with the numbers 18, 12, and 30. Just write each number down in its own row at the top of the grid.
  2. 2
    Write the lowest common prime factor of the numbers in the space to the left. Just look out for the smallest prime factor (such as 2, 3, or 5) that you can pull out of all the numbers. They're all even, so you can pull out 2.
  3. 3
    Divide each of the original numbers by the common prime factor. Write the quotient below each number. Here's how to do it:
    • 18/2 = 9, so write 9 below 18.
    • 12/2 = 6, so write 6 below 12.
    • 30/2 = 15, so write 15 below 30.
  4. 4
    Repeat the process of pulling out and dividing by the lowest prime factor until no more common factors exist. Just repeat the process from the previous steps using the numbers 9, 6, and 15 this time.
  5. 5
    If two of the numbers still share a prime common factor, then continue the process until no pair of bottom numbers have a common factor. In this particular example, you're done.
    • For example, if the bottom three numbers are 2, 39, and 122, divide 2 and 122 by 2 leaving the new bottom row as 1, 39, and 61.
  6. 6
    Multiply all the numbers of the first column containing the common prime factors with the numbers at the bottoms of all the other columns. This is the LCM. In this example, the product of the common factor column is 6 (2 x 3). Multiply 6 by the numbers at the bottoms of the other columns: 6 x 3 x 2 x 5 = 180.

Method 4
Use Euclid's Algorithm

  1. 1
    Use Euclid's Algorithm to find the Greatest Common Divisor (GCD) of two numbers. Let's say the two numbers you're using are 210 and 45. Here's how you use Euclid's Algorithm to find the GCD of both numbers:[2]
  2. 2
    Multiply the two original numbers. 210 x 45 = 9,450
  3. 3
    Divide the result by the GCD of the two numbers. 9,450/15 = 630. 630 is the LCM of 210 and 45.
  4. 4
    Use Euclid's Algorithm to find the LCM of three numbers. To do this, simply find the GCD of two numbers, and then use that GCD to find the LCM of those two numbers and the third.

Community Q&A

Add New Question
  • What is the least common multiple of 105, 210 and 630?
    wikiHow Contributor
    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.
  • Why in the first method is 42 used instead of 2?
    wikiHow Contributor
    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.
  • 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.
    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.
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  • If you want to know if the LCM will be less than or greater than the product, use this method: If their GCF is 1, their LCM is their product. If their GCF is greater than 1, their LCM will be less than their product.
  • If you need to find the LCM of more than two numbers, the above method needs to be tweaked, because it only works for two numbers at a time. For instance, to find the LCM of 16, 20, and 32, we could start by finding the LCM of 16 and 20 (which, as we said, is 80), and then find the LCM of 80 and 32, which turns out to be 160.
  • For instance, to find the LCM of 16 and 20, we take the G.C.D of 16 and 20, which comes out to 4. 16 × 20 = 320, and 320 ÷ 4 = 80, so 80 is the LCM.
  • If you need to convert a fraction into a common denominator, you will need to know how many times each denominator goes into the LCM. When you use this method, you can find the conversion factor by multiplying the numbers at the bottom of all the other columns (excluding the first one listing common prime factors). So to convert 18 into 180, multiply by 2 and 5. To convert 12 into 180, multiply by 3 and 5. To convert 30 into 180, multiply by 3 and 2.
  • 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. For instance, to add 1/6 + 3/8, we find the LCM of 6 and 8, which is 24, and convert each fraction to have a denominator of 24, which changes the problem to 4/24 + 9/24. Then we can just add the numerator, which gives us 13/24.
  • Suppose the given numbers are 306 and 657 and their HCF( highest common factor) is 9....to find the LCM we multiply the numbers 306×657=201,042 and then we divide their product with the HCF=9 we get 201042÷9=22338....and we get the LCM = 22338....

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Categories: Mathematics

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Español: encontrar el mínimo común múltiplo de dos números, Português: Encontrar o Menor Múltiplo Comum de Dois Números, Italiano: Trovare il Minimo Comune Multiplo di Due Numeri, Français: trouver le plus petit commun multiple de deux nombres, Deutsch: Kleinstes gemeinsames Vielfaches von zwei Zahlen berechnen, Русский: найти наименьшее общее кратное двух чисел, 中文: 求两个数的最小公倍数, Nederlands: Het kleinste gemene veelvoud van twee getallen vinden, Bahasa Indonesia: Mencari Kelipatan Perkalian Terkecil dari Dua Angka

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