FINDING THE GREATEST COMMON FACTOR OF [beautiful math coming... please be patient] $\,2\,$ or $\,3\,$ NUMBERS
EXAMPLE:
Question: Find the greatest common factor of [beautiful math coming... please be patient] $\,27\,$ and $\,18\,$:
Answer: [beautiful math coming... please be patient] $\,9$
The idea:
The factors of $\,27\,$ are: $\,1\,$   $\,3\,$   $\,9\,$   $\,27$
The factors of [beautiful math coming... please be patient] $\,18\,$ are: $\,1\,$ $\,2\,$ $\,3\,$ $\,6\,$ $\,9\,$ $\,18\,$  
The common factors of [beautiful math coming... please be patient] $\,27\,$ and $\,18\,$ (the numbers that appear in both lists) are $\,1\,$, $\,3\,$, and $\,9\,$.
The greatest common factor (the greatest number in the list of common factors) is $\,9\,$.
EFFICIENT ALGORITHM FOR FINDING THE GREATEST COMMON FACTOR

Here's an efficient algorithm for finding the greatest common factor,
when there aren't too many numbers, and they aren't too big.

The process is illustrated by finding the greatest common factor of [beautiful math coming... please be patient] $\,18\,$, $\,36\,$, and $\,90\,$:

  • Line up the numbers in a row. (See the purple rectangle at left.)
  • Find ANY number that goes into everything evenly (like $\,2\,$).
  • Do the divisions, and write the results above the original numbers.
    In the example:
    [beautiful math coming... please be patient] $18\,$ divided by $\,2\,$ is $\,9\,$; write the $\,9\,$ above the $\,18\,$
    [beautiful math coming... please be patient] $36\,$ divided by $\,2\,$ is $\,18\,$; write the $\,18\,$ above the $\,36\,$, and so on.
  • Keep repeating the process, until there isn't any number (except $\,1\,$) that goes into everything evenly.
  • Multiply the circled numbers together. This is the greatest common factor!
  • In the example, [beautiful math coming... please be patient] $\,\text{gcf}(18,36,90) = 2\cdot 3\cdot 3 = 18\,$.
WHY DOES THIS METHOD WORK?

Think about why this method works.
As you walk through each step of this discussion, keep comparing with the chart above.
Look at the prime factorizations of each number:

[beautiful math coming... please be patient] $18 = 2\cdot3\cdot 3$
[beautiful math coming... please be patient] $36 = 2\cdot 2\cdot 3\cdot 3$
[beautiful math coming... please be patient] $90 = 2\cdot3\cdot 3\cdot 5$
The number $\,2\,$ goes into each evenly, so separate it off:
[beautiful math coming... please be patient] $18 = 2\cdot (3\cdot 3)\hphantom{\cdot 3} = 2\cdot 9$
[beautiful math coming... please be patient] $36 = 2\cdot (2\cdot 3\cdot 3) = 2\cdot 18$
[beautiful math coming... please be patient] $90 = 2\cdot (3\cdot 3\cdot 5) = 2\cdot 45$
The number $\,3\,$ goes into each remaining part evenly, so separate it off:
[beautiful math coming... please be patient] $18 = (2\cdot 3) \cdot (3) \hphantom{\cdot 3}= (2\cdot 3)\cdot 3$
[beautiful math coming... please be patient] $36 = (2\cdot 3) \cdot (2\cdot 3) = (2\cdot 3)\cdot 6$
[beautiful math coming... please be patient] $90 = (2\cdot 3) \cdot (3\cdot 5) = (2\cdot 3)\cdot 15$
Here's the final step:
[beautiful math coming... please be patient] $18 = (2\cdot 3\cdot 3)\cdot 1$
[beautiful math coming... please be patient] $36 = (2\cdot 3\cdot 3)\cdot 2$
[beautiful math coming... please be patient] $90 = (2\cdot 3\cdot 3)\cdot 5$
OTHER WAYS TO APPLY THE ALGORITHM

Here are some other ways the algorithm might be applied.
Of course, you get the same answer any correct way that you do it!

You can also zip over to wolframalpha.com and type in:   gcd(18,36,90)
The ‘gcd’ stands for greatest common divisor.

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Finding the Greatest Common Factor of Variable Expressions

 
 
Find the greatest common factor of: