Prime Numbers

Example 1

2 is a prime number, why?

2 can be divided by 1     2 ÷ 1 = 2 with remainder equal to zero

2 can be divided by 2 (itself)     2 ÷ 2 = 1 with remainder equal to zero

Try to find another whole number that divides 2 with remainder zero. There is not.

Example 2

7 is a prime number, why?

7 can be divided by 1     7 ÷ 1 = 7 with remainder equal to zero

7 can be divided by 7 (itself)     7 ÷ 7 = 1 with remainder equal to zero

Try to find another whole number that divides 7 with remainder zero. There is not.

The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

You can generate more prime numbers and test them.

Composite Numbers

Definition: Any whole number that can be divided (with remainder equal to zero) by another whole number other than 1 and itself is called composite number.

4 is a composite number, why?

4 can be divided by 1, itself and 2.

6 is a composite number, why?

6 can be divided by 1, itself, 2 and 3.

12 is a composite number : it can be divided by 1, itself, 2, 3, 4 and 6.

30 is a composite number: it can be divided by 1, itself, 2, 3, 5, 6, 10 and 15.

Factorization

From division to multiplication to factoring.

Division and multiplication are related operations.

The division 6 ÷ 3 = 2 may be written as a multiplication: 6 = 2 × 3

Examples

1) 6 = 1 × 6 ; 6 = 2 × 3     1, 2, 3 and 6 are called factors of 6.

2) 12 = 12 ×1 = 3 ×4 = 6 ×2     1, 2, 3, 4, 6 and 12 are called factors of 12.

3) 20 = 1 × 20 = 2 ×10 = 2 × 2 × 5 = 4 × 5     1, 2, 3, 4, 5, 10 and 20 are called factors of 20.

Prime factorization

Examples

1) 6 = 2 × 3     the factors 2 and 3 are prime numbers.

2) 12 = 2 × 2 × 3     the factors 2 and 3 are prime numbers.

3) 20 = 2 × 2 × 5     the factors 2 and 5 are prime numbers.

Example 1

Write the prime factorization of 12

1) See if the first prime number 2 is a factor of the given number 12

12 ÷ 2 = 6 with remainder = 0        2 is a factor of 12        12 = 2 × 6

2) See if the first prime number 2 is a factor of 6

6 ÷ 2 = 3 with remainder = 0        2 is a factor of 6        6 = 2 × 3        Hence 12 = 2 × 6 = 2 × 2 × 3

12 = 2 × 2 × 3 is completely factored using only prime numbers 2 and 3

Example 2

Write the prime factorization of 21

1) See if the first prime number 2 is a factor of the given number 21

21 ÷ 2 = 10 but remainder = 1 so 2 is not a factor of 21

2) Is the next prime number 3 a factor of 21?

21 ÷ 3 = 7 with remainder 0        3 is a factor of 21        21 = 3 × 7

3 and 7 are prime numbers and therefore 21 = 3 × 7 is completely factored using only prime numbers 3 and 7

This prime factor calculator can be used to generate all prime factors of a given number.

Answer the Following Questions

1. For each number below, decide which is a prime and which is a composite number? Explain.
   
   1. 9

   2. 11

   3. 16

   4. 22

   5. 39

   6. 41

   7. 49

   8. 57
2. Which of the following is a prime factorization?
   
   1. 8 = 2 × 4

   2. 10 = 2 × 5

   3. 20 = 2 × 10

   4. 30 = 2 × 3 × 5

   5. 38 = 2 × 19

   6. 42 = 2 × 3 × 7

   7. 56 = 2 × 2 × 14

   8. 75 = 3 ×25

   9. 80 = 2 × 2 × 2 × 2 × 5

   10. 100 = 2 × 2 × 25
3. What is the prime factorization to each of the following whole numbers?
   
   1. 8

   2. 18

   3. 24

   4. 45

   5. 63

   6. 88

   7. 96

Solutions to the Above Problems

1. 
   
   A prime number has only 2 factors: 1 and itself. A composite has 3 factors or more.
   1. 9 is a composite number since 9 = 3 × 3 , it has 3 fators: 1, 3 and itself

   2. 11 is a prime number because it has only 2 factors: 1 and itself

   3. 16 is a composite number since 16 = 4 × 4 = 2 × 8 , it has 5 fators: 1, 2, 4 , 8 and itself

   4. 22 is a composite number since 22 = 2 × 11 , it has 4 fators: 1, 2, 11 and itself

   5. 39 is a composite number since 39 = 3 × 13 , it has 4 factors : 1, 3, 13 and itself

   6. 41 is a prime number because it has only 2 factors: 1 and itself

   7. 49 is a composite number since 49 = 7 × 7 , it has 4 factors : 1, 7 and itself

   8. 57 is a composite number since 57 = 3 × 19 , it has 4 factors: 1, 3, 19 and itself
2. 
   A prime factorization includes prime numbers only.
   1. 8 = 2 × 4           the factor 4 is not a prime number therefore it is not a prime factrorization.

   2. 10 = 2 × 5           both factors 2 and 5 are prime numbers so it is a prime factrorization.

   3. 20 = 2 × 10           the factor 10 is not a prime number therefore it is not a prime factrorization.

   4. 30 = 2 × 3 × 5 ;           all factors 2, 3 and 5 are prime numbers so it is a prime factrorization.

   5. 38 = 2 × 19           both factors 2 and 19 are prime numbers so it is a prime factrorization.

   6. 42 = 2 × 3 × 7           all factors 2, 3 and 7 are prime numbers so it is a prime factrorization.

   7. 56 = 2 × 2 × 14           the factor 14 is not a prime number therefore it is not a prime factrorization.

   8. 75 = 3 × 25           the factor 25 is not a prime number therefore it is not a prime factrorization.

   9. 80 = 2 × 2 × 2 × 2 × 5           all factors 2 and 5 are prime numbers so it is a prime factrorization.

   10. 100 = 4 × 25           both factors are composite; it is not a prime factrorization.
3. 
   Prime factorization is carried out by successive divions using prime numbers.
   1. 8 = 2 × 4 = 2 × 2 × 2

   2. 18 = 2 × 9 = 2 × 3 × 3

   3. 24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3

   4. 45 = 3 × 15 = 3 × 3 × 5

   5. 63 = 7 × 9 = 7 × 3 × 3

   6. 88 = 2 × 44 = 2 × 2 × 22 = 2 × 2 × 2 × 11

   7. 96 = 2 × 48 = 2 × 2 × 24 = 2 × 2 × 2 × 12 = 2 × 2 × 2 × 2 × 6 = 2 × 2 × 2 × 2 × 2 × 3

References