# What are all the greatest common factors of 36 and 90?

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Jul 21, 2017

$G C F = 18$
Common factors:$\text{ } 1 , 2 , 3 , 6 , 9 , 18$

#### Explanation:

There can be several common factors, but there is only one Greatest Common factor.

Write 36 and 90 as the product of their prime factors.

$36 = 2 \times 2 \times 3 \times 3$

$90 = \textcolor{w h i t e}{\times x} 2 \times 3 \times 3 \times 5$

$G C F = \textcolor{w h i t e}{x} 2 \times 3 \times 3 \textcolor{w h i t e}{\times x} = 18$

As for all the common factors, it is probably easiest to write all the factors of 36 and then select which are factors of 90 as well.

Factors of 36: $\text{ "color(red)(1, 2, 3), 4, " "color(red)(6, 9)," " 12," " color(red)(18), " } 36$
Factors of 90$\text{ "color(red)(1,2,3)" ",5,color(red)(6,9),10," "15,color(red)(18),30," } 45 , 90$

Common factors:$\text{ } \textcolor{red}{1 , 2 , 3 , 6 , 9 , 18}$

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EET-AP Share
Oct 9, 2017

There is only one greatest common factor of 36 and 90 which is 18.

There are also a number of common factors including 1, 2, 3, 6, 9, 18.

#### Explanation:

What is the greatest common factor (GCF)?
That is the largest number that will divide into all those given.
To find it, the smallest prime numbers should be divided into each one. Prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19.

For the given numbers $36$ and $90$, both divided by $2$ give $18$ and $45$.

$18$ will divide into both $36$ and $90$, but $45$ will not, so $18$ is the GCF.

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Oct 14, 2017

G C F 18
It’s also called Greatest Common Divisor G C D

#### Explanation:

To find G C F of 36, 90:

First write the factors of both terms :
Factors of $36 = 2 \cdot \textcolor{red}{2 \cdot 3 \cdot 3}$

Factors of $90 = \textcolor{red}{2 \cdot 3 \cdot 3 \cdot} 5$

Select the common factors in both terms as marked redabove.
$\textcolor{red}{2 \cdot 3 \cdot 3} =$18 is the G C F

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Nov 21, 2017

Here's a way to find the GCF without using prime factors

#### Explanation:

Instead of finding the prime factors of the two numbers,
~ make a list of ALL the factors of each number
~ then pick the largest ("greatest") one they have in common.

To find ALL the factors of a number:
~ Start by factoring by 1 and writing the factors down.
~ Then factor by 2, then by 3, then by 4, and so on.
~ If a number will not go in evenly, it is not a factor, so skip it and go to the next number.
~ When the factor pairs start to repeat, you are done.

The factors of 36
1 $\times$ 36
2 $\times$ 18
3 $\times$ 12
4 $\times$ 9
5 $\times$ $\leftarrow$ not a factor, so skip to 6
6 $\times$ 6 $\leftarrow$ The factors will now just repeat, so you are done.
The factors of 36 are:
1, 2, 3, 4, 6, 9, 12, $\textcolor{red}{18}$, 36

Now compare those factors to the factors of 90
The factors of 90
1 $\times$ 90
2 $\times$ 45
3 $\times$ 30
4 $\times$ $\leftarrow$ Not a factor, so skip to 5
5 $\times$ 18
6 $\times$ 15
7 $\times$ $\leftarrow$ Skip
8 $\times$ $\leftarrow$ Skip
9 $\times$ 10
10$\times$ 9 $\leftarrow$ The factors are repeating now, so the list is complete
The factors of 90 are:
1, 2, 3, 5, 6, 9, 10, 15, $\textcolor{red}{18}$, 30, 45, 90
........................................

The factors that 36 and 90 have in common are:
1, 2, 3, 6, 9, 18
So 18 is the greatest common factor
.........................................

This technique of listing all possible factors (instead of prime factors) comes in handy for various applications.
For one thing, there's no chance you will miss a factor.

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