Question

How do you find the greatest common factor of 88, 66?

Answer 1

`GCF(66,88)=22`

Explanation:

list all the factors of the two numbers then find the common ones and pick the largest from the list

factors 66:`{1,2,3,6,11,22,33,66}`
factors 88:`{1,2,4,8,11,22,44,88}`

common factors:`{1,2,3,6,11,22,33,66}nn{1,2,4,8,11,22,44,88}`

`{1,2,11,22}`

`GCF(66,88)=22`

Answer 2

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

Explanation:

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"`.