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

Oct 31, 2016

$G C F \left(66 , 88\right) = 22$

#### Explanation:

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

factors 66:$\left\{1 , 2 , 3 , 6 , 11 , 22 , 33 , 66\right\}$
factors 88:$\left\{1 , 2 , 4 , 8 , 11 , 22 , 44 , 88\right\}$

common factors:$\left\{1 , 2 , 3 , 6 , 11 , 22 , 33 , 66\right\} \cap \left\{1 , 2 , 4 , 8 , 11 , 22 , 44 , 88\right\}$

$\left\{1 , 2 , 11 , 22\right\}$

$G C F \left(66 , 88\right) = 22$

Nov 1, 2016

$G C F \left(88 , 66\right) = 22$, but there is another way to calculate it, sometimes more usefull...

#### Explanation:

Calculate the $G C F$ 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 $G C F$ fairly quickly, whatever numbers we have to consider, and the strategy used also serves to other operations and related integer calculations (eg , calculating the $L C M$, simplifying radicals or fractions ...).

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

$\textcolor{w h i t e}{\text{0000}}$For example, suppose you want to find the $G C F$ of $600$, $1500$
$\textcolor{w h i t e}{\text{0000}}$and $3300$. The factorization of these three numbers is:

$\textcolor{w h i t e}{\text{00000000000000000000}} 600 = {2}^{3} \cdot 3 \cdot {5}^{2}$
$\textcolor{w h i t e}{\text{0000000000000000000}} 1500 = {2}^{2} \cdot 3 \cdot {5}^{3}$
$\textcolor{w h i t e}{\text{0000000000000000000}} 3300 = {2}^{2} \cdot 3 \cdot {5}^{2} \cdot 11$

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

$\textcolor{w h i t e}{\text{0000}}$In our example, as the powers with equal bases on the three
$\textcolor{w h i t e}{\text{0000}}$numbers are those with base 2, 3 and 5, those would be the
$\textcolor{w h i t e}{\text{0000}}$factors to consider. The factor 11, however, only appears in the
$\textcolor{w h i t e}{\text{0000}}$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.

$\textcolor{w h i t e}{\text{0000}}$Of all the factors that have $2$ as a base, the smallest exponent
$\textcolor{w h i t e}{\text{0000}}$that appears is the $2$, therefore, we will use ${2}^{2}$ in calculating the
$\textcolor{w h i t e}{\text{0000}} G C F$. We do the same with the $3$ (which is raised to $1$ in the
$\textcolor{w h i t e}{\text{0000}}$three numbers, so we'll use ${3}^{1}$) and $5$ (which has the smallest
$\textcolor{w h i t e}{\text{0000}}$exponent $2$):

$\textcolor{w h i t e}{\text{000000000000}} G C F \left(600 , 1500 , 3300\right) = {2}^{2} \cdot 3 \cdot {5}^{2} = 300$.

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

$\textcolor{w h i t e}{\text{0000" GCF = "common factors with lower exponent}}$.