What Is The Greatest Common Factor? (5 Key Concepts)


When factoring expressions in algebra, it is sometimes helpful to find the greatest common factor of two or more numbers.  The greatest common factor also has applications in number theory and cryptography.

So, what is the greatest common factor?  The greatest common factor (also called greatest common divisor) of a set of integers is the largest positive whole number that is a factor of each integer in the set. For a set of two numbers, we can use prime factorizations or a list of multiples to find the greatest common factor (GCF).

Of course, the greatest common factor is related to the least common multiple for a set of numbers, and there is a formula that shows this relationship.

In this article, we’ll talk about the greatest common factor of a set of numbers and how to find it.  We’ll also look at some examples to make the concept clear.

Let’s get started.

What Is The Greatest Common Factor?

The greatest common factor of a set of integers is the largest positive whole number that is a factor of each integer in the set.  The greatest common factor (GCF) is also called the greatest common divisor (GCD).

greatest common factor of 210 and 66
The greatest common factor (GCF) of 210 and 66 is 6, since 6 is the largest positive whole number that is a factor of both 210 and 66. You can see the prime factors for each number (and the shared prime factors) in this Venn Diagram.

Note that the set of integers has two or more numbers, and not all of them are zero.  For the set of numbers {x1, x2, … , xn}, the greatest common factor is GCF(x1, x2, … , xn) or GCD(x1, x2, … , xn).

If d is the greatest common factor of the set of numbers {x1, x2, … , xn}, then we can write the following equations:

  • x1 = dy1
  • x2 = dy2
  • xn = dyn

where y1, y2, … , yn are integers.  These equations are just telling us that d is a factor of each number in the set.

We can also find the greatest common factor of a set of two numbers.  If a is not zero, then the greatest common factor of 0 and a is |a|.  That is:

  • GCF(0, a) = GCF(a, 0) = |a|

The greatest common factor of a set of numbers is related to another concept: the least common multiple.

Relationship Between Greatest Common Factor & Least Common Multiple

The least common multiple of a set of integers is the smallest positive whole number that is a multiple of each integer in the set.  The least common multiple (LCM)

If m is the least common multiple of the set of numbers {x1, x2, … , xn}, then we can write the following equations:

  • m = x1y1
  • m = x2y2
  • m = xnyn

where y1, y2, … , yn are integers.  These equations are just telling us that m is a multiple of each number in the set.

There is a key equation that relates the greatest common factor and least common multiple of a set of numbers.  Here is the equation for a set of two numbers a and b:

  • LCM(a, b)*GCF(a, b) = |a*b|

To say it another way: first, calculate the least common multiple (LCM) of a and b.  Next, calculate the greatest common factor (GCF) of a and b.  Then, take the product of the LCM and GCF.  This will be the same as the absolute value of the product of a and b.

Now let’s see how to find the greatest common factor in practice.

How To Find The Greatest Common Factor

There are two methods you can use to find the greatest common factor of two numbers:

  • One method is to make a list of the integer multiples for each number.  Then, find the smallest number that shows up in all of the lists – this is the GCF.
  • Another method is to use prime factorization of each number to find the least common multiple of the numbers and then use the formula above to find the GCF.

Let’s look at some examples where we solve with both methods.

Example 1: Finding The Greatest Common Factor Of Two Numbers (Method 1)

Let’s say we want to find the greatest common factor (GCF) of the numbers 12 and 32, using the first method mentioned above.

First, we make a list of the integer multiples of each number (multiples of 12 and multiples of 32):

  • Integer multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, …
  • Integer multiples of 32: 32, 64, 96, 128, 160, 192, 224, 256, …

Now, we look for the smallest number that shows up in both lists.  It looks like 96 is the smallest number that shows up in both lists (96 = 12*8 and 96 = 32*3).

So, the least common multiple (LCM) of 12 and 32 is 96.

Finally, we use the equation relating GCF and LCM to find the GCF:

  • LCM(a, b)*GCF(a, b) = |a*b|
  • LCM(12, 32)*GCF(12, 32) = |12*32|  [a = 12, b = 32]
  • 96*GCF(12, 32) = 384
  • GCF(12, 32) = 384/96
  • GCF(12, 32) = 4

So, the greatest common factor of 12 and 32 is 4.

We can verify this calculation by using the second method (shown below).

Example 2: Finding The Greatest Common Factor Of Two Numbers (Method 2)

Let’s say we want to find the greatest common factor (GCF) of the numbers 12 and 32, using the second method mentioned above.

First, we find the complete prime factorization of each number:

  • Prime Factorization of 12: 12 = 22*3
  • Prime Factorization of 32: 32 = 25

Next, we find the prime factors that appear in both factorizations.  Here, only the prime factor of 2 shows up in both factorizations.

Then, for each prime factor, we take the lowest power of that prime factor that appears in any factorization.  Here, the powers of 2 are 2 and 5 (22 for 12 and 25 for 32).  So, we take an exponent of 2 for 2, giving us a factor of 22.

Finally, we multiply all of the primes raised to powers to give us our GCF.  Here, we only have 22, so we get 4 for the GCF.

This matches what we found in the previous example using the other method.  We can use the same formula as before to find the LCM of 12 and 32.

Example 3: Finding The Greatest Common Factor Of Three Numbers (Method 2)

Let’s say we want to find the greatest common factor (GCF) of the numbers 36, 54, and 90, using the second method mentioned above.

First, we find the complete prime factorization of each number:

  • Prime Factorization of 36: 36 = 22*32
  • Prime Factorization of 54: 54 = 2*33
  • Prime Factorization of 90: 90 = 2*32*5

Next, we find the prime factors that appear in both factorizations.  Here, the prime factors 2 and 3 both show up in all three factorizations.

Then, for each prime factor, we take the lowest power of that prime factor that appears in any factorization.  Here, the lowest power of 2 is 1 (for 54 and 90), while the lowest power of 3 is 2 (for 12 and 90).

So, we take an exponent of 1 for 2 and an exponent of 2 for 3, giving us factors of 21 and 32.  Taking the product, we get 21*32 = 2*9 = 18, which is the GCF.

So, the greatest common factor of 36, 54, and 90 is 18.

Can A Greatest Common Factor Be Prime?

A greatest common factor can be a prime number in some cases.  However, this is not always true.

Example 1: A Greatest Common Factor That Is Prime

Let’s find the greatest common factor of 40 and 45.

  • Prime Factorization of 40: 40 = 23*5
  • Prime Factorization of 45: 45 = 32*5

The only prime factor that appears in both factorizations is 5, and the lowest power of 5 is 51.  So, the GCF of 40 and 45 is 5, which is a prime number.

Example 2: A Greatest Common Factor That Is Not Prime

Let’s find the greatest common factor of 40 and 45.

  • Prime Factorization of 80: 80 = 24*5
  • Prime Factorization of 90: 90 = 2*32*5

The prime factors that appear in both factorizations are 2 and 5, and the lowest powers are 21 and 51.  So, the product 2*5 = 10 gives us the GCF of 80 and 90.

Since 10 is not prime, the GCF of 80 and 90 is not prime.

Can 1 Be A Greatest Common Factor?

The greatest common factor of two numbers can be 1 if they have no common prime factors.

Example: Greatest Common Factor Of 1

Let’s find the greatest common factor of 110 and 273.

  • Prime Factorization of 110: 110 = 2*5*11
  • Prime Factorization of 273: 273 = 3*7*13

There is no prime number that shows up in both factorizations.  So, we are just taking a product of the number 1 itself, which means the GCF of 110 and 273 is 1.

Can A Number Be Its Own Greatest Common Factor?

A number can be its own greatest common factor if it is a multiple of the second number.

Example: A Number That Is Its Own Greatest Common Factor

Let’s find the greatest common factor of 10 and 70.

  • Prime Factorization of 10: 10 = 2*5
  • Prime Factorization of 70: 70 = 2*5*7

The prime factors that appear in both factorizations are 2 and 5, and the lowest powers are 21 and 51.  So, the product 2*5 = 10 gives us the GCF of 10 and 70.

Since 10 is a factor of 70 (70 = 10*7), the GCF is equal to the first number in the set (which is 10).

Can A Greatest Common Factor Be A Decimal?

A greatest common factor cannot be a decimal.  By definition, the greatest common factor is a positive integer (whole number).

This means that a greatest common factor will not contain any decimals or fractions.

Can A Greatest Common Factor Be Negative?

A greatest common factor cannot be negative.  By definition, the greatest common factor is a positive integer (whole number).

Conclusion

Now you know about the greatest common factor of a set of numbers and how to find it.  You also know the answers to some common questions about the greatest common factor.

You can learn about perfect numbers and what they are in this article.

I hope you found this article helpful.  If so, please share it with someone who can use the information.

Don’t forget to subscribe to my YouTube channel & get updates on new math videos!

~Jonathon

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