#GCF (88, 66) = 22#, but there is another way to calculate it, sometimes more usefull...

Calculate the #GCF# making a list of the factors of numbers and looking for the biggest of those who are repeated is a simple method but it can be very slow to use if we have more than two numbers and they are of a large size.

Instead, using the other method I describe below, you can calculate the #GCF# fairly quickly, whatever numbers we have to consider, and the strategy used also serves to other operations and related integer calculations (eg , calculating the #LCM#, simplifying radicals or fractions ...).

(1) For each of the numbers that we have to consider, we make its prime factorization:

#color(white) "0000"#For example, suppose you want to find the #GCF# of #600#, #1500#
#color(white) "0000"#and #3300#. The factorization of these three numbers is:

#color(white) "00000000000000000000" 600 = 2^3 cdot 3 cdot 5^2#
#color(white) "0000000000000000000" 1500 = 2^2 cdot 3 cdot 5^3#
#color(white) "0000000000000000000" 3300 = 2^2 cdot 3 cdot 5^2 cdot 11#

(2) We chose those factors that are repeated in all numbers, first taking the base of each.

#color(white) "0000"#In our example, as the powers with equal bases on the three
#color(white) "0000"#numbers are those with base 2, 3 and 5, those would be the
#color(white) "0000"#factors to consider. The factor 11, however, only appears in the
#color(white) "0000"#decomposition of one of the numbers, so we discard it:

#color(white) "000000000000" GCF (600, 1500, 3300) = 2^? cdot 3^? cdot 5^?#

(3) We must use as exponents, for each base, the smallest of which appear in the prime factorization.

#color(white) "0000"#Of all the factors that have #2# as a base, the smallest exponent
#color(white) "0000"#that appears is the #2#, therefore, we will use #2^2# in calculating the
#color(white) "0000" GCF#. We do the same with the #3# (which is raised to #1# in the
#color(white) "0000"#three numbers, so we'll use #3^1#) and #5# (which has the smallest
#color(white) "0000"#exponent #2#):

#color(white) "000000000000" GCF (600, 1500, 3300) = 2^2 cdot 3 cdot 5^2 = 300#.

We can recall the method of calculating the #GCF# learning that "we take only those factors that are repeated, and using the smallest possible exponent". More abbreviated form:

#color(white) "0000" GCF = "common factors with lower exponent"#.