**How To Find The GCF And LCM**

In this article we will learn to find the Greatest Common Factor (GCF) and Least (lowest) Common multiple (LCM) of integers. In mathematics, especially when dealing with fractions, you come across multiple situations where you need to find either the GCF or LCM. It is a very useful skill to master and let’s see what this is all about.

Suppose your mom has two cakes for a party. The vanilla flavored cake is smaller and has an area of 12 square inches and the strawberry flavored cake is larger with an area of 20 square inches. Your mom wants to divide both cakes into pieces such that all pieces are of equal size and both cakes are evenly divided with no leftover smaller pieces.

Tricky situation! What should the size of pieces be to achieve that? Perhaps you could help your mom if you know about the GCF! GCF is the greatest common factor, which means that the greatest possible number by which both numbers are divisible.

*The GCF of 12 and 20 is 4!*

The numbers 12 and 20 are both divisible by 4 and there is no integer greater than 4 by which both 12 and 20 are divisible. This means that if you cut the pieces such that their area is 4 square inches, then both cakes will be evenly divided into equal portion sizes. But, how do you find the GCF? Read on to find out!

**Prime Factorization**

First thing to learn before everything else is how to do prime factorization. I am sure you are already familiar with the concept of prime numbers. A prime number is a number that is not divisible by any number except by itself and 1, for example, 2, 3, 5, 11, 13, 17 e.t.c. Notice that there are no two numbers (except 1 and the number itself) which, when multiplied together, give these numbers as an answer.

Prime factorization of a number means that you factor that number into prime factors only. In other words, you determine the prime numbers, which when multiplied give the original number. There are multiple methods to visualize and remember how to do that, but we will outline the one that is most common and easy to remember. Whatever the visualization method is, the underlying procedure is the same:

*Keep on dividing the number by prime numbers until the answer is 1*

Start with the smallest prime number, if the number is not divisible by that move on to the next (i.e. start with 2, if 2 is not a factor, try 3, if 3 is not a factor, try 5 and so on).

Let’s find the prime factors of 12:

- Divide 12 by 2 and the answer is 6
- Divide the answer 6 again by prime number 2 and now the answer is 3
- Divide the latest answer 3 again by prime number. Since it is not divisible by 2, we use 3. 3 divided by 3 is 1!
- The division stops when we get 1 as final answer
- Now list all the prime numbers used in division as factors of 12:
**2 x 2 x 3**

The procedure outlined is illustrated below for clarity:

1.

2. Now divide 6 by prime number:

3. Now divide 3 by prime number

4. Division stops when you get 1 at the end.

5. List the prime factors:

Remember this visualization technique to find prime factors of a number. Now let’s find the prime factors of 20.

*Prime Factors of 20 = 2 x 2 x 5*

Sometimes, we write 1 in the factors as well. Writing 1 does not affect our answer. So, the prime factors of 12 and 20 are written as:

**Primer Factors of 12 = 1 x 2 x 2 x 3**

**Prime Factors of 20 = 1 x 2 x 2 x 5**

It is desirable to write 1 in the factors because it is sometimes useful when finding the GCF. We will learn how below.

**Finding GCF: **

That was all about finding the prime factors. Now we will see how prime factorization helps in finding the GCF. Follow these following simple steps to find the GCF:

Find the GCF of 12 and 20

**Step 1**: List prime factors of both numbers

**Step 2**: Identify the numbers that appear in the factors of both numbers. (Remember that the integers you select and their repetitions must be common in the factors of both. For example, if 2 appears 4 times in the factors of one number and 3 times in the factors of the other number, you select integer 2 with 3 repetitions, because 2 appears 3 times in the factors of both). For the example we are solving:

**Step 3**: Multiply those factors identified in step two to get the GCF:

**1 × 2 × 2 = 4**

Here we go! The GCF of 12 and 20 is 4. Now you can confidently tell your mom the size of cake pieces she needs to cut in order to achieve her goal.

Let’s do another example:

Find the GCF of 15 and 13

**Step 1**: Prime factorization:

*Prime factorization of 15:*

Prime factorization of 13:

**Step 2**: Here you see there are no common factors except 1. Recall that we mentioned earlier that writing 1 in the factors is important for finding the GCF sometimes. Now you can see that when no common factors exist, 1 is always a common factor.

**Step 3**: So, the GCF is 1!

Similarly, you can find the GCF of more than two numbers. Find the prime factors of all the numbers and then identify the common factors. Multiply the common factors to get the GCF of the numbers!

That was all about GCF, so now we will look into the LCM.

**Finding LCM:**

The least common multiple of two or more numbers, is a number which is the smallest number divisible by all the numbers. Finding the LCM also involves finding prime factors. Let’s do some examples.

Find the LCM of 16, 20 and 30

Follow these steps to find the LCM:

**Step 1:** List prime factors of the numbers

**Step 2:** Write the factors in index form i.e. when a factor occurs more than once, write it raised to some power. For example, when 2 is multiplied 4 times it can be written as 2 raised to the 4th power: \(2^{4}\). When a number has no power, it means its power is 1.

**Step 3:** Choose factors. If a factor occurs in case of only one number it is always chosen, if a factor occurs in both numbers, its maximum power is chosen.

That was all about the LCM.

**Final Thoughts!**

The LCM and GCF are important concepts that are used extensively in mathematics. For example, addition or subtraction of fractions involves finding the LCM of denominators and simplifying the fractions involves finding the GCF of the numerator and denominator. Similarly, there are many other instances which require computation of the LCM and GCF and you can confidently master this skill by going through this article.