# What are the factors of 90?

Mar 7, 2018

See a solution process below:

#### Explanation:

First, because 90 is an even number we can factor out a 2:

$90 = 2 \times 45$

Because 45 ends in a 5 it is divisible by 5 so we can factor out a 5:

$90 = 2 \times 5 \times 9$

We know 9 is divisible by 3 so we can factor our a 3:

$90 = 2 \times 5 \times 3 \times 3$

or

$90 = 2 \times 3 \times 3 \times 5$

Mar 7, 2018

$1 , 2 , 3 , 5 , 6 , 9 , 10 , 15 , 18 , 30 , 45 , 90$

#### Explanation:

When dealing with factors, you can list all the factors, or write a number as the product of the prime factors.

If you know the product of the prime factors you can use those to determine all the other factors.

$90 = 10 \times 9$

$\therefore 90 = 2 \times 5 \times 3 \times 3 \text{ } = 2 \times 3 \times 3 \times 5$

Listing the factors using these prime factors can be done as follows:

$1$ at a time: $1 , 2 , 3 , 5$
Product of $2$ prime factors at a time: $6 , 10 , 9 , 15$
Product of $3$ prime factors at a time: $18 , 30 , 45$
Product of $4$ prime factors at a time: $90$

The factors work in pairs, a big one with a small one.

Half of the factors are smaller than $\sqrt{90}$ and half are bigger.
$\sqrt{90} = 9. \ldots .$

In order they are:

$1 , \text{ "2," "3," "5," "6," "9," "10," "15," "18," "30," "45," } 90$
$\textcolor{w h i t e}{\times \times \times \times \mathcal{\mathcal{\times}} \times . \times \times x} \uparrow$
$\textcolor{w h i t e}{\times \times \times \times \times \times , \ldots . . x} \sqrt{90}$

Apr 26, 2018

$\left\{1 , 2 , 3 , 5 , 6 , 9 , 10 , 15 , 18 , 30 , 45 , 90\right\}$

#### Explanation:

One way is to find pairs of number that multiply to $90$ it is important though to be systematic so that you ensure you find them all.

$90 = 1 \times 90$

$90 = 2 \times 45$

$90 = 3 \times 30$

$90 = 5 \times 18$

$90 = 6 \times 15$

$90 = 9 \times 10$

$90 = 10 \times 9$

and the we repeat

so factors

$\left\{1 , 2 , 3 , 5 , 6 , 9 , 10 , 15 , 18 , 30 , 45 , 90\right\}$