Finding the greatest common factor of a pair or set of numbers
Defining common factors and greatest common factors
A common factor of two positive whole numbers is a number that divides evenly into both of them.
Their greatest common factor (sometimes abbreviated GCF) is the largest number that divides evenly into both of them.
Another name that’s used for “greatest common factor” is “greatest common divisor” (sometimes abbreviated GCD).
One way to find the greatest common factor of two positive whole numbers is to find the prime factorizations of both numbers and then to look for the factors that appear in both factorizations.
Using a prime factorization to find the greatest common factor of a set of values
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Finding the greatest common factor of 14 and 28
Example
Find the greatest common factor of ???14??? and ???28???.
In order to find the greatest common factor, we need to look for the largest number that divides evenly into both ???14??? and ???28???.
To do this, we’ll break down number into its prime factors.
The factor of ???2??? appears once in ???14??? and twice in ???28???, so we’ll have one factor of ???2??? in the greatest common factor. The factor of ???7??? appears once in ???14??? and once in ???28???, so we’ll have one factor of ???7??? in the greatest common factor.
Therefore, the greatest common factor of ???14??? and ???28??? is ???2\cdot7???. Multiplying this out, we get ???2\cdot7=14???, which means ???14??? is the greatest common factor of ???14??? and ???28???.
Let’s double-check our answer by making sure that ???14??? divides evenly into both ???14??? and ???28???.
???14\div14=1???
???28\div14=2???
It does, so ???14??? is a common factor of ???14??? and ???28???. In fact, ???14??? is the greatest common factor of ???14??? and ???28???: When we divide ???14??? and ???28??? by ???14???, we get ???1??? and ???2???, respectively, and the only number that divides evenly into both ???1??? and ???2??? is ???1???.
In that example, the greatest common factor ???14??? was equal to one of the original numbers, but that won’t always be the case.