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 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 ...

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

Example 1

Factor 4 into prime numbers

Solution

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:

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:

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:

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:

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:

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

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

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