Least Common Denominator

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What is a “Multiple” ?

The multiples of a number are what you get when you multiply it by other numbers (such as if you multiply it by 1,2,3,4,5, etc). Just like the multiplication table.

Here are some examples:

 The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, etc … The multiples of 12 are: 12, 24, 36, 48, 60, 72, etc…

The least common multiple (LCM) of two or more numbers is the smallest number that is divisible by each of the numbers.

There are two widely used methods.

Method 1: Simply list the multiples of each number, then look for the smallest number that appears in each list.

Example: Find the least common multiple for 5, 6, and 15.

• First we list the multiples of each number.

Multiples of 5 are 10, 15, 20, 25, 30, 35, 40,…

Multiples of 6 are 12, 18, 24, 30, 36, 42, 48,…

Multiples of 15 are 30, 45, 60, 75, 90,….

• Now, when you look at the list of multiples, you can see that 30 is the smallest number that appears in each list.
• Therefore, the least common multiple of 5, 6 and 15 is 30.

Method 2: To use this method factor each of the numbers into primes. Then for each different prime number in all of the factorizations, do the following…

1. Factor into primes.
2. For each prime number, take the largest of these counts.
3. Write down that prime number as many times as you counted for it in step 2.
4. The least common multiple is the product of all the prime numbers written down.

Example: Find the least common multiple of 5, 6 and 15.

Step #1: Factor into primes

Prime factorization of 5 is 5

Prime factorization of 6 is 2 3

Prime factorization of 75 is

Step #2: For each prime number, take the one with the largest power.

There is only one 2 so we take

There are two 3s having equal powers so we take any of them

There are two 5s having unequal powers so we take the one with the largest power which is

Step #3 – The least common multiple is the product of all the prime numbers written down.

2 x 3 x 5 = 30

Therefore, the least common multiple of 5, 6 and 15 is 30.

So there you have it. A quick and easy method for finding least common multiples.

What is a Denominator?

The denominator is the bottom number in a fraction.

It shows how many equal parts the item is divided into

What is a Common Denominator?

Common Denominator” just means that the denominators

in two (or more) fractions are the same.

Why is it Important?

Before you can add or subtract fractions, the fractions need to have a common denominator (in other words the denominators must be the same). If the denominators are not the same, you can use the Least Common Multiple to make them the same:

Here is an example:

You can’t add fractions with different denominators:

So what do you do? How can they be added?

Answer: You need to make the denominators the same by finding the (LCM) of the denominators.

Step #1: Factor the denominators into primes

Prime factorization of 3 is 3

Prime factorization of 6 is 2 3

Step #2: For each prime number, take the one with the largest power.

There is only one 2 so we take

There are two 3s having equal powers so we take any of them

Step #3 – The least common multiple is the product of all the prime numbers written down.

2 x 3 = 6

Therefore, the least common multiple of 3 and 6 is 6.

Now change each fraction (using LCM) to make their denominators the same as the (LCM)

We need to change the the first fraction only

So 1/3 is multiplied up and down by 2, and it becomes 2/6

The pizzas now look like this (thus the fractions can be added)

Fruit shoot (LCD)

Snow ball fight (LCM)