Greatest Common Factors

What is a GCF?

GCF stands for Greatest Common Factor. You need at least two whole numbers to find the greatest common factor, because the word common here means that we are looking for a factor that two or more numbers share. Since 1 is a factor of every number, any two or more numbers have a GCF, because even if the numbers are prime, or if they don’t share any prime factors, then the GCF will be 1. If they do share any prime factors, then the GCF will be the product of those shared prime factors.

How do you find the Greatest Common Factor of two numbers?

There’s more than one way to find the GCF of two numbers.

Technique #1: Find the GCF by listing all factors — obvious, and easy to remember, but possibly a lot of work.

  1. Make a list of every factor of the first number.
  2. Make a list of every factor of the second number.
  3. Scan the lists — the GCF is the largest number that is on both lists.
  4. In order to adapt this technique for the GCF of three or more numbers, just make lists of every factor for each number. The GCF is the largest number that is present on every one of the lists.

Although listing all the factors is a straightforward approach, it can be far more work than necessary. The amount of work to find the GCF can be greatly reduced using prime number factorization.

Technique #2: Find the GCF using prime number factorization.

  1. Find the prime factorization of the first number.
  2. Find the prime factorization of the second number.
  3. Make a list of all the primes that are present in both lists. If multiple copies of primes are present, include only as many copies as are present in both numbers.
  4. Multiply together all the common primes on the list made in step 3. The product is the GCF.
  5. In order to adapt this technique for the GCF of 3 or more numbers, find the prime factorization of each number, and then in step 3 collect factors that are present in all the lists.

The reason that Technique #2 is less work is that as each number is broken into smaller and smaller factors as part of the prime factorization steps, the problem gets easier and easier, whereas with Technique #1, one has to keep working with the full size number until all potential factors are checked to see if they are really factors.

What are Greatest Common Factors used for?

Perhaps the most common use of the GCF is when simplifying a fraction or cross-canceling while multiplying fractions. However, in cases where the GCF can’t be easily seen, and has to be found by factorization, there is usually no reason to multiply the prime factors out to find the GCF since they can simply be cancelled in place, saving work.

Sometimes the GCF is used to simplify expressions by “bringing it out” via the distributive property.

What is the difference between the GCF and the LCM?

The Greatest Common Factor and the Least Common Multiple are similar ideas, but in a sense, they are opposites. Both have to do with the prime factorization of the two or more numbers in question. The difference is what is done after the prime factorization of the numbers is found. The GCF is the product of only the prime factors that are found in each and every one of the numbers. The LCM is the product of all the prime factors that are found in either of the numbers.

What’s the difference between the GCF and the LCD?

An LCD (Least Common Denominator) is just the LCM (Least Common Multiple) of two denominators. So this question is really similar to the previous one about GCFs and LCMs. However, the GCF is a bit more generic in that it can refer to any two whole numbers, whereas the LCD is the LCM of two numbers that are specifically found in denominators.

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