Finding the least common multiple, also known as the lowest common denominator, of two or more numbers is a fairly simple process. Once you understand the basics, you will be able to find the least common multiple of any numbers!
To find the least common multiple of two or more numbers, you will need to know how to divide each number by every other number. For example, to find the least common multiple of 3 and 4, you would have to know how to divide each number by 2 and 1, respectively.
Using this knowledge, you can now write the numerator and denominator of a fraction with all factors in common. Fractionating the numerator and denominator like this will give you a constant value that is left over when dividing one number from another.
Remove the greatest common factor from both numbers
Now let’s try a more difficult example. Find the least common multiple of 3-2+1 and 2-1. First, remove the greatest common factor of 2 from both numbers.
Now we have to figure out how to combine these two numbers. We need to find a way to add and subtract them so they both end up as 1.
The addition and subtraction trick is to write out the number that is 1 higher or lower than the original numbers, then add and subtract them. In this case, we will write 1+1 and 1-1, then add and subtract them.
The least common multiple of 3-2+1 and 2-1 is 3*3=9. The least common multiple of 3-2+1 and 2-1 is 9.
Find the least common multiple of both numbers
The least common multiple (LCM) of two numbers x and y is the smallest positive number that is a multiple of both x and y.
For example, the LCM of 6 and 8 is 24, since 24 is the smallest positive number that is a multiple of both 6 and 8.
How to find LCM? You can use the formula: LCM(a, b) = (a * b) / gcd(a, b), where gcd(a, b) is the greatest common divisor of a and b.
In our case we need to find the least common multiple of 3 and -1. The greatest common divisor of 3 and -1 is 1, so the least common multiple of 3 and -1 is 1×1=1.
Write the answer in factored form
The last thing you can do with prime factors is write the number in factored form. When you divide a number by a prime number, you get another number that is called the quotient.
When you divide a number by 1, you get the same number but with no numbers after it. This is called the zero principle, or a null principle.
For example, if you divide 12 by 2, you get 6 as the quotient. If you divide 12 by 1, then you get 12 as the same number without any numbers after it.
When writing numbers in factored form, write all of the prime factors before writing the 1 that makes up the whole number. Write these in alphabetical order to make it easier to identify them.
The last step is to re-write the whole number as a product of its prime factors. Check your work to make sure they are all correct and in alphabetical order.
Example with numbers
We will start with an example using numbers. Let’s say you need to find the least common multiple of 3-2+1 and 2-1. First, find the prime factors of each number.
3-2+1 = 1*3=1*2=2*1=3
2-1 = 2
Then, find the prime factors of the least common multiple you need to find. In this case, you need a multiple of both 3 and 2, so those are the prime factors you need to look for. You also need a multiple of 1, so that is the last prime factor you need to look for.
Now that you have found all of the prime factors, put them in order and write out all of the possible products. For example, take the first two prime factors: 1 and 2. There are two possible products: 1×2 or 2×1.
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