20 Jun ##GCF (88 66) = 22## but there is another way to calculate i
##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 : ##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 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##.
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