# What is the prime factorization of 18 and 20?

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To get the prime factorization of a number, apply divisibility rules.

(a) 18

Since 18 is an even number, it is divisible by the smallest even number which is 2 (`18-:2=6` ). So,

`18 = 2 xx 9`

Then, factor 9 further. Since 9 is an odd number and a multiple of 3, then it is divisible by 3 (`9-:3=3` ).

`18 =2xx3xx3`

Now that factors are all prime numbers, express the repeated factors in exponent form.

**Therefore, `18=2xx3^2` .**

(b) 20

20 is an even number. So,applying the divisibility rule for even numbers, its factor will be:

`20=2xx10`

10 is an even number too. So it can be factor further by applying the same rule.

`20=2xx2xx5`

Now that the factors are all prime numbers, express the repeated factors in exponent form.

**Thus, `20=2^2xx5` .**

Prime numbers are numbers that only have factors of 1 and that number. For example, 2 can only be factored to be 2x1, therefore it is a prime number. If you want the prime factorization of 18 and 20, you could use a factor tree or just do it in your head.

18 can be simplified to 6x3. 3 is already a prime number, so now you just split 6 into 2x3. **2x3x3** is your prime factorization for 18. Easy, right?

For 20, you can simplify it to 4x5. 5 is a prime number, so now you can split 4 into 2x2. Your prime factorization for 20 is **2x2x5**. For larger numbers, a factor tree would be the best method, but since these are small numbers you can do it easily.

The factors of 18 are 1, 2, 3, 6, 9, 18

The prime factors of 18 are 2*3*3 so 18 = 2*3*3

The factors of 20 are 1, 2, 4, 5, 10, 20

The prime factors of 20 are 5*2*2 so 20 = 5*2*2

Prime factorization could always be solved using a factor tree. No matter how big or how small the numbers are. Attached in a picture below are the factor trees for 18 and 20.

For 18, the prime numbers are 2, 3, 3

therefore 18 = 2 * 3 * 3

For 20, the prime numbers are 5, 2, 2

therefore 20 = 5 * 2 * 2

For smaller numbers like 18,20, you can list out all the factors and get the prime ones.

For larger numbers, the easiest way is to just keep dividing by small prime numbers until you reach a prime number. Basically, you divide until the answer is prime. (Remember that the final prime answer is also part of the factorization)

Start this process by dividing by 2. Always. Only when the number is not even, should you move onto trying other numbers.

A few tricks to know what number is divisible by what are

- If the digits of the number add up to something that is divisble by 3, then that number is also divisible by 3. (Ex. 27 -> 2+7=9 and 9/3=3 so 3 is a factor of 27)

- If the number ends in 0 or 5 it's divisible by 5

So for 18,

18/2=9

9 is odd, so try 3. 9/3=3, and 3 is prime, so

18=2x3x3

For 20,

20/2=10, 10/2=5. 5 is prime, and you are done.

20=2x2x5

The Factors of 18 are : 1, 2, 3, 6, 9, 18

The prime factor of 18=2x3x3

Factor of 20:

The Factors of 20 are : 1, 2, 4, 5, 10, 20

the prime factor of 20=2x2x5

18 (Pick a number that goes into 18 evenly, such as 2.)

9 x 2 (Realize that 9 can be factored further still.)

3 x 3 x 2 (This is your answer and can be written as (3^2) x 2

20 (Pick a number that goes into 20 evenly, such as 5)

5 x 4 (What number can be further factored still?)

5 x 2 x 2 (This is your answer and can be written as (2^2) x 5