# Prime Factoring of a Whole Numbers

Examples on how to find the prime factors of a positive integer are presented. You may also use prime factors online calculator to factor positive integers and check your answers. More questions and their solutions are included.

## Factors

Factors of a positive integer n are all the positive integers that divide n with remainder equal to zero.
For example 2, 3, 4 and 6 are factors of 12 because
12 ÷ 2 = 6,     with remainder equal to zero.
12 ÷ 3 = 4,     with remainder equal to zero.
12 ÷ 4 = 3     with remainder equal to zero.
and     12 ÷ 6 = 2 with remainder equal to zero.
These factors are also present when 12 is written as a product of factors (factoring).
For example     12 = 6 × 2,     12 = 4 × 3 ...

## Prime Factoring

The fundamental theorem of arithmetic states that there is only one way that a given positive integer can be represented as a product of one or more prime numbers.

A Prime number n is a positive integer greater than 1 that has only 1 and n (itself) as positive integer divisors.
Below is a list of the first prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,...

How to find prime factors of a positive integer?
We start by dividing the given number by 2 if possible, then divide the result by 2 if possible; if not divide by the next prime number 3 and continue until the quotient is a prime number

### Examples

Example 1
Factor 4 into prime numbers
Solution

4 = 2 . 2 = 2 2

Example 2
Factor 20 into prime numbers
Solution
step 1.    20 = 2 × 10
step 2. Factor 10 and write    20 = 2 × 2 × 5
Since 5 is a prime number, we cannot factor further. Hence the prime factoring of 20 is given below:

20 = 2 × 2 × 5 = 2 2 × 5

Example 3
Factor 100 into prime numbers
Solution
step 1.    100 = 2 × 50
step2. Factor 50 and write    100 = 2 × 2 × 25
step3. Factor 25 and write    100 = 2 × 2 × 5 × 5
5 is a prime number and therefore we cannot factor it. Hence the prime factoring of 100 is given as:

100 = 2 × 2 × 5 × 5 = 2 2 × 5 2

Example 4
Factor 1020 into prime numbers
Solution
step 1.    1020 = 2 × 510
step 2.    1020 = 2 × 2 × 255
step 3.    1020 = 2 × 2 × 3 × 85
step 4.    1020 = 2 × 2 × 3 × 5 × 17
17 is a prime number and therefore we cannot factor it. Hence the prime factoring of 1020 is given as:

1020 = 2 × 2 × 3 × 5 × 17 = 2 2 × 3 × 5 × 17

Example 5
Factor 634 into prime numbers
Solution
step 1.    634 = 2 × 317
317 is a prime number and therefore we cannot factor it. Hence the prime factoring of 634 is given as:

634 = 2 × 317

Example 6
Factor 720 into prime numbers
Solution
step 1.    720 = 2 × 360
step 2. Factor 360 and write:   720 = 2 × 2 × 180
step 3. Factor 180 and write:    720 = 2 × 2 × 2 × 90
step 4. Factor 90 and write:   720 = 2 × 2 × 2 × 2 × 45
step 5. Factor 45 and write:    720 = 2 × 2 × 2 × 2 × 3 × 15
step 6. Factor 15 and write:    720 = 2 × 2 × 2 × 2 × 3 × 3 × 5
5 is a prime number and therefore we cannot factor it. Hence the prime factoring of 720 is given as:

720 = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 2 4 × 3 2 × 5

## Questions

(with solutions)

Part A
Which of the following does not represent a prime factorization?
a) 2 × 2 × 4 ,       b) 3 × 3 × 5 × 9 ,       c) 3 × 3 × 5 × 17 ,       d) 2 × 5 × 5 × 21 ,       e) 2 × 2 × 3 × 5 × 41

Part B
Factor the following numbers into prime numbers
a) 18 ,       b) 300 ,       c) 123 ,       d) 1200 ,       e) 1450

## Solutions to the Above Questions

Part A
a) 2 × 2 × 4    is NOT factored in prime numbers because the factor 4 is not a prime number.
b) 3 × 3 × 5 × 9    is NOT factored in prime numbers because the factor 9 is not a prime number.
c) 3 × 3 × 5 × 17    all factors included are prime numbers and therefore   3 × 3 × 5 × 17   represents a prime factorization
d) 2 × 5 × 5 × 21    is NOT factored in prime numbers because the factor 21 is not a prime number.
e) All factors included are prime numbers and therefore   2 × 2 × 3 × 5 × 41   represents a prime factorization

Part B
a) 18 = 2 × 9
= 2 × 3 × 3 × 3 = 2 × 3 2

b) 300 = 2 × 150
= 2 × 2 × 75
= 2 × 2 × 3 × 25
= 2 × 2 × 3 × 5 × 5 = 2 2 × 3 × 5 2

c) 123 = 3 × 41

d) 1200 = 2 × 600
= 2 × 2 × 300
= 2 × 2 × 2 × 150
= 2 × 2 × 2 × 2 × 75
= 2 × 2 × 2 × 2 × 3 × 25
= 2 × 2 × 2 × 2 × 3 × 5 × 5 = 2 4 × 3 × 5 2

e) 1450 = 2 × 725
= 2 × 5 × 145
= 2 × 5 × 5 × 29 = 2 × 5 2 × 29