Greatest Common Divisor: Definition & Formula

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  • 0:00 Greatest Common Divisor
  • 1:14 Prime Factorization Method
  • 2:35 Euclid's Algorithm Method
  • 4:51 Lesson Summary
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Lesson Transcript
Instructor: Kimberly Osborn
The greatest common divisor, also known as the greatest common factor, can be found through a variety of methods. This lesson will focus on both the prime factorization method and Euclid's algorithm.

Greatest Common Divisor

A while ago, I decided to throw a party for the Super Bowl. When looking through my supplies, I realized that I had 56 chicken wings that I could cook and 32 cans of soda. I wanted to invite as many friends as possible to my party but I wanted each person to get an equal number of chicken wings and cans of soda. Here was my dilemma. How was I going to figure out the largest number of friends that I could invite to my party? My dilemma was a classic example of needing to find the greatest common divisor among two or more numbers.

The greatest common divisor is simply the biggest number that can go into two or more numbers without leaving a remainder, or the biggest factor that the numbers share. In the case of my Super Bowl party, I wanted to find the largest number of friends that could I invite so that each person got an equal number of chicken wings and cans of soda.

You might encounter other versions of the greatest common divisor, including greatest common factor, highest common divisor, and highest common factor. In this video, we'll be looking at two methods that can be used to find the greatest common divisor: the prime factorization method and Euclid's algorithm method.

Prime Factorization Method

Let's start by using some prime factorization trees to find all of the prime factors shared by two or more numbers; in this case, 56 and 32. Here's two prime factorization trees for each number.

Prime Factorization Tree

Remember, when creating a prime factorization tree, start with the original number and create branches at each level with two quotients that equal the number above it until you are only left with prime numbers. In the case of 32, our prime factorization tree branches out until it reaches 2 x 2 x 2 x 2 x 2. In the case of 56, our prime factorization tree branches out until it reaches 2 x 2 x 2 x 7.

Make sure to order your prime factorizations from smallest to largest so that it is easier to see the numbers they have in common. Placing our prime factorizations on top of each other, we can now see exactly what prime numbers they share. These numbers are circled in red.

Lined up factors for 32 and 56

The last step in this method for finding the greatest common divisor is to multiply these common numbers together. So, in the case of my Super Bowl party, I can invite 2 x 2 x 2, or 8 friends. This means that the largest number of friends that I can invite to my party so that each person gets an equal number of chicken wings and cans of soda is 8.

Euclid's Algorithm Method

If you aren't a fan of the prime factorization method for finding the greatest common divisor, you can also use Euclid's algorithm to reach the same solution.

The equation for Euclid's algorithm is:

Euclid

Before jumping into how this method works, there are a few important vocabulary words that we need to understand from the algorithm.

  • Dividend: the number being divided
  • Divisor: The number doing the dividing
  • Quotient: The number being multiplied by the divisor to equal the dividend
  • Remainder: The amount left over after the divisor and quotient are multiplied

That's a lot of new vocabulary, but bear with me. I promise it will all make sense by the end of this lesson.

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