Show Steps for Working Out by: None Listing Multiples Prime Factorization Cake / Ladder Division Method GCF Method
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The Least Common Multiple (LCM) is also referred to as the Lowest Common Multiple (LCM) and Least Common Divisor (LCD). For two integers a and b, denoted LCM(a,b), the LCM is the smallest positive integer that is evenly divisible by both a and b. For example, LCM(2,3) = 6 and LCM(6,10) = 30.

The LCM of two or more numbers is the smallest number that is evenly divisible by all numbers in the set.

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Least Common Multiple Calculator

Find the LCM of a set of numbers with this calculator which also shows the steps and how to do the work.

Input the numbers you want to find the LCM for. You can use commas or spaces to separate your numbers. But do not use commas within your numbers. For example, enter 2500, 1000 and not 2,500, 1,000.

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How to Find the Least Common Multiple LCM

This LCM calculator with steps finds the LCM and shows the work using 5 different methods:

Listing Multiples Prime Factorization Cake/Ladder Method Division Method Using the Greatest Common Factor GCF

How to Find LCM by Listing Multiples

List the multiples of each number until at least one of the multiples appears on all lists Find the smallest number that is on all of the lists This number is the LCM

Example: LCM(6,7,21)

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 Multiples of 7: 7, 14, 21, 28, 35, 42, 56, 63 Multiples of 21: 21, 42, 63 Find the smallest number that is on all of the lists. We have it in bold above. So LCM(6, 7, 21) is 42

How to find LCM by Prime Factorization

Find all the prime factors of each given number. List all the prime numbers found, as many times as they occur most often for any one given number. Multiply the list of prime factors together to find the LCM.

The LCM(a,b) is calculated by finding the prime factorization of both a and b. Use the same process for the LCM of more than 2 numbers.

For example, for LCM(12,30) we find:

Prime factorization of 12 = 2 × 2 × 3 Prime factorization of 30 = 2 × 3 × 5 Using all prime numbers found as often as each occurs most often we take 2 × 2 × 3 × 5 = 60 Therefore LCM(12,30) = 60.

For example, for LCM(24,300) we find:

Prime factorization of 24 = 2 × 2 × 2 × 3 Prime factorization of 300 = 2 × 2 × 3 × 5 × 5 Using all prime numbers found as often as each occurs most often we take 2 × 2 × 2 × 3 × 5 × 5 = 600 Therefore LCM(24,300) = 600.

How to find LCM by Prime Factorization using Exponents

Find all the prime factors of each given number and write them in exponent form. List all the prime numbers found, using the highest exponent found for each. Multiply the list of prime factors with exponents together to find the LCM.

Example: LCM(12,18,30)

Prime factors of 12 = 2 × 2 × 3 = 22 × 31 Prime factors of 18 = 2 × 3 × 3 = 21 × 32 Prime factors of 30 = 2 × 3 × 5 = 21 × 31 × 51 List all the prime numbers found, as many times as they occur most often for any one given number and multiply them together to find the LCM 2 × 2 × 3 × 3 × 5 = 180 Using exponents instead, multiply together each of the prime numbers with the highest power 22 × 32 × 51 = 180 So LCM(12,18,30) = 180

Example: LCM(24,300)

Prime factors of 24 = 2 × 2 × 2 × 3 = 23 × 31 Prime factors of 300 = 2 × 2 × 3 × 5 × 5 = 22 × 31 × 52 List all the prime numbers found, as many times as they occur most often for any one given number and multiply them together to find the LCM 2 × 2 × 2 × 3 × 5 × 5 = 600 Using exponents instead, multiply together each of the prime numbers with the highest power 23 × 31 × 52 = 600 So LCM(24,300) = 600

How to Find LCM Using the Cake Method (Ladder Method)

The cake method uses division to find the LCM of a set of numbers. People use the cake or ladder method as the fastest and easiest way to find the LCM because it is simple division.

The cake method is the same as the ladder method, the box method, the factor box method and the grid method of shortcuts to find the LCM. The boxes and grids might look a little different, but they all use division by primes to find LCM.