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:
0000For example, suppose you want to find the GCF of 600, 1500
0000and 3300. The factorization of these three numbers is:
00000000000000000000600=23⋅3⋅52
00000000000000000001500=22⋅3⋅53
00000000000000000003300=22⋅3⋅52⋅11
(2) We chose those factors that are repeated in all numbers, first taking the base of each.
0000In our example, as the powers with equal bases on the three
0000numbers are those with base 2, 3 and 5, those would be the
0000factors to consider. The factor 11, however, only appears in the
0000decomposition of one of the numbers, so we discard it:
000000000000GCF(600,1500,3300)=2?⋅3?⋅5?
(3) We must use as exponents, for each base, the smallest of which appear in the prime factorization.
0000Of all the factors that have 2 as a base, the smallest exponent
0000that appears is the 2, therefore, we will use 22 in calculating the
0000GCF. We do the same with the 3 (which is raised to 1 in the
0000three numbers, so we'll use 31) and 5 (which has the smallest
0000exponent 2):
000000000000GCF(600,1500,3300)=22⋅3⋅52=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:
0000GCF=common factors with lower exponent.
