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In order to find the greatest common factor of monomials follow these two steps.

Step 1. Find the prime factorization of each monomial.

Step 2. Take the product of all common factors

Example of finding the GCF of a monomial.

Find the greatest common factor of 
27x^2yq^5 and 15x^3q^3

Step 1. Find the prime factorization of each monomial
27x^2yq^5 = 3 3 3 x x  y  q q q q q
15x^3q^3 =  3  5  x x x  q q q

Step 2. Take the product of common factors

How is Least Common Multiple different than Greatest Common Factor?

The least common multiple is the smallest positive integer that is a multiple of two or more integers.

For example the least common multiple of 4 and 16 is 16

The GCF is the greatest integer that divides evenly into two or more of integers.

The GCF of 4 and 16 is 4

Finding the greatest common factor of monomials

video tutorial greatest common factor
finding lcm
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finding lcm

Example 2 Finding the Greatest Common Factor (GCF)

Greatest Common Factor
Welcome to MooMooMath. Today we are going to talk about the greatest common factor. Here are our first examples in fast form. Let’s look at 30 the first thing I’m going to factor is 5 times 6, the 5 is prime so I don’t bring it down, next 6 equals it 3 three times 2 so I write a 5 times 2 times 3. Now let’s do a factor three of 42, that 6 times 7, 6 is composite and 7 is prime so I’m done with that one so now I can factor the 6 down to 2 times 3 so the prime factorization is 2 times 3 times 7, so I circle my common factors and multiply so my greatest common factor is 6. Now let’s slow it down and look at the rules of greatest common factor.
1. Complete prime factorization for each number. This includes drawing a factor tree for each number and factor down to prime numbers.
2. Pair the common factoring. ( It may help to circle all the factors that are common )
3. Take the common factors (the ones that you circled) and multiple them back together and this gives you your GCF.
So let’s take it slowly with our steps in an example. The first step is beginning our prime factorization. The last time I took 5 times 6 but this time I’m going to factor it differently I’m going to do 2 times 15 just to show you a different method. OK 2 is prime, so I’m finished with that stem so I draw another stem off the 15 and it factors into 5 times 3. Notice that I end up with the same prime factorization that I had before. It is 2 times 5 times 3. Now let’s go over here and complete the prime factorization for 42. Since 42 is an even number I will write 2 down. That is 2 times 21 and the 21 factors into 3 times 7 so those are my three factors 2 times 3 times 7. So this completes step one, I have completed my prime factorization. Now I will circle the common factors. Both 30 and 42 have 2 as a common factor and both 30 and 42 have three as a common factor so all I do is multiply 2 times 3 and that gives me my GCFof 6 Hope this was helpful.

How to find the GCF

Greatest common factor using a factor tree
Finding the GCF can become confusing and frustrating if the only method you use is trial and error. Here is one proven method for finding the GCF.
Steps for finding greatest common factor or GCF picture has lady standing by a dry erase board with steps written down
1. Set up factor tree
2. Factor down to prime numbers I like to circle or underline prime numbers
3.Multiply common factors 3x5 = 15 is the GCF
Steps for finding greatest common factor or GCF picture has lady standing by a dry erase board with steps written down

Greatest Common Factor Definition

For example, the greatest common factor of 24 and 36 is 12 because it is the largest positive integer that will divide  into both numbers without a remainder.
24/12 = 2     36/12 = 3

Integers are positive or negative whole numbers. Integers do not contain fractions

  • Whole numbers greater than 0 are positive integers
  • Whole numbers less than 0 are negative integers
  • Zero is an integer and is considered "neutral"
The GCF or Greatest Common Factor is simply the largest positive integer that will divide exactly into two or more numbers.