Question

What is the prime factorization of 180?

Answer 1

`180=2^2xx3^2xx5`

Explanation:

We have to obtain the factors of 180 which are `color(blue)"prime numbers"`

A`color(blue)" prime number"` being a number with only 2 factors.That is 1 and the number itself.

For example `"5 only has 2 factors 1 and 5"rArr5=1xx5`

Here are the first few `color(blue)"prime numbers"`

`2,3,5,7,11,13,.....`

A number which is not prime is a `color(red)"composite number".` Composite numbers have more than 2 factors. All composite numbers can be expressed as a product of prime factors.

For example 10 has factors 1 ,2 ,5 and 10 but can be expressed as a product of prime factors.

`rArr10=2xx5`

There are a variety of ways to find the prime factors of a number. Here is one.

Starting with 180, divide it by the lowest prime number 2 , repeat until it cannot be divided by 2 and move on to next prime number 3 and so on. When the remainder is 1 stop-we have obtained our prime factors.

`•180÷color(red)(2)=90`

`•90÷color(red)(2)=45 " which cannot be divided by" 2to3`

`•45÷color(red)(3)=15`

`•15÷color(red)(3)=5" which cannot be divided by" 3to5`

`•5÷color(red)(5)=1" having reached 1, we are there"`

`rArr180=color(red)(2xx2xx3xx3xx5)=2^2xx3^2xx5`