Find the Least Common Multiple (Lcm) of Two Numbers

The Least Common Multiple (or Lowest Common Multiple) of a group of numbers, called the LCM, is the smallest number that's a multiple of all the numbers. For instance, the LCM of 16 and 20 is 80; 80 is the smallest number that's both a multiple of 16 and a multiple of 20.

There are several methods of finding the LCM of two numbers, but one of the easiest is this one. Let's call the two numbers 'a' and 'b'.

[edit] Using Euclid's Algorithm

Let's call our two numbers "A" and "B".

 Use Euclid's Algorithm to find the Greatest Common Divisor (GCD) of 'A' and 'B'. 
 Multiply A × B.
Divide the result of  step 2 by the GCD you found in the step 1. The result is the LCM!

This second method is slower and require factoring the two numbers, but is trivially extensible to handle many numbers at once, and also produces the conversion factor if the LCM is to be used to add fractions.

[edit] Tips for using this method

If you need the LCM of three numbers, begin by finding the LCM of the first two. Then take that LCM and the third number, and find the LCM of those two. For example, to find the LCM of 2, 3, and 5, start with LCM(2, 3). That gets you 6. Then find LCM(6, 5) to get 30.

[edit] Using a Common Factors Grid

Write the numbers in a row, leaving a small space on the left and as much space as possible below.

Figure out the lowest common prime factor, and write it in the space to the left.
Divide each number in the row by that common prime factor, writing the quotient below each.
Repeat until no more common prime factors exist. Now the first column is a list of common prime factors.
Multiply the number at the top of the second column with the numbers at the bottoms of the other columns. This is the LCM.

As an added bonus, the list of factors in the first column, when multiplied, produce the GCF (greatest common factor).

[edit] Tips for using this method

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.

[edit] Using the Prime Factorization Method

Label the two numbers "a" and "b".
Break each number down into its list of prime factors. (For more info, see the article How to Factor a Number.)
Circle the common factors (numbers that show up on both lists in step 2).
Underline the non-common factors on list "a" and put a squiggle under the non-common factors on list "b".
- At this point, every factor should be either circled,underlined, or squiggled.
Make a list of the numbers from step 3, followed by the numbers from step 4.
Multiply everything on the list together to get one big number. This is the LCM.

[edit] Tips for using this method

If you need the LCM in order to convert a fraction into a common denominator, you will need to know what to multiply each fraction with in order to create the LCM. The conversion factor for "a" will be all the numbers with squiggles under them on list "b". The conversion for "b" will be all the numbers underlined on list "a".

[edit] Using the Guess-and-Check Method

Start with the larger of the two numbers for which you want the LCM. Label it "a". Label the smaller number "b".
Write down the first four or five multiples of "a". (In other words, take a x 2, then a x 3, then a x 4, etc.)
Go through the list, from smallest to largest, dividing each multiple by "b" until you find one that doesn't have a remainder.
If you don't find one, then extend the list (a x 5, a x 6, a x 7, etc.) and repeat the process.

[edit] Tips

For instance, to find the LCM of 16 and 20, we take the GCD of 16 and 20, which comes out to 4. 16 × 20 = 320, and 320 ÷ 4 = 80, so 80 is the LCM.
The LCM has a lot of uses. The most common is that, whenever you add or subtract fractions, they must have the same denominator; if they don't, you need to convert each fraction to some equivalent fraction so that they'll 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 numerators, which gives us 13/24.

[edit] Warnings

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.