Prime Factorization

Back to Algebra–Ready Review Content.

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

Prime Factorization

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y

Prime Factorization

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factor of x.

Prime Factorization

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factor of x. Hence 15 is a multiple of 3 (or 5),

Prime Factorization

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factor of x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15.

Prime Factorization

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factor of x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1.

Prime Factorization

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factor of x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1.

A prime number is a natural number that can only be divided by 1 and itself.

Prime Factorization

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factor of x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1.

A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only.

Prime Factorization

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factor of x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1.

A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers.

Prime Factorization

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factor of x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1.

A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers. 15 is not a prime number because 15 can be divided by 3 or 5.

Prime Factorization

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factor of x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1.

A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers. 15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number.

Prime Factorization

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factor of x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1.

A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers. 15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number.

Example A. List the factors and the multiples of 12.

Prime Factorization

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factor of x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1.

A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers. 15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number.

Example A. List the factors and the multiples of 12.

The factors of 12:The multiples of 12:

Prime Factorization

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factor of x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1.

A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers. 15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number.

Example A. List the factors and the multiples of 12.

The factors of 12: 1, 2, 3, 4, 6, and 12.The multiples of 12:

Prime Factorization

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factor of x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1.

A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers. 15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number.

Example A. List the factors and the multiples of 12.

The factors of 12: 1, 2, 3, 4, 6, and 12.The multiples of 12: 12=1x12,

Prime Factorization

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factor of x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1.

A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers. 15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number.

Example A. List the factors and the multiples of 12.

The factors of 12: 1, 2, 3, 4, 6, and 12.The multiples of 12: 12=1x12, 24=2x12,

Prime Factorization

Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items.

If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y or y is a factor of x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. Note that a number x is always the multiple of itself and the number 1.

A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. Hence 2, 3, 5, 7, 11, 13,… are prime numbers. 15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number.

Example A. List the factors and the multiples of 12.

The factors of 12: 1, 2, 3, 4, 6, and 12.The multiples of 12: 12=1x12, 24=2x12, 36=3x12, 48, 60, etc…

Prime Factorization

To factor a number x means to write x as a product, i.e. write x as a*b for some a and b.

Prime Factorization

To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as

12 = 2*6 = 3*4 = 2*2*3.

Prime Factorization

To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as

12 = 2*6 = 3*4 = 2*2*3.

We say the factorization is complete if all the factors are prime numbers.

Prime Factorization

To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as

12 = 2*6 = 3*4 = 2*2*3.

We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factored completely,

Prime Factorization

To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as

12 = 2*6 = 3*4 = 2*2*3.

We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factored completely, 2*6 is not complete because 6 is not a prime number.

Prime Factorization

To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as

12 = 2*6 = 3*4 = 2*2*3.

We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factored completely, 2*6 is not complete because 6 is not a prime number.

Exponents

Prime Factorization

To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as

12 = 2*6 = 3*4 = 2*2*3.

We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factored completely, 2*6 is not complete because 6 is not a prime number.

Exponents

To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.

Prime Factorization

To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as

12 = 2*6 = 3*4 = 2*2*3.

We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factored completely, 2*6 is not complete because 6 is not a prime number.

Exponents

To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.

We write x*x*x…*x as xN where N is the number of x’s multiplied to itself.

Prime Factorization

To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as

12 = 2*6 = 3*4 = 2*2*3.

We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factored completely, 2*6 is not complete because 6 is not a prime number.

Exponents

To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.

We write x*x*x…*x as xN where N is the number of x’s multiplied to itself. N is called the exponent, or the power of x.

Prime Factorization

To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as

12 = 2*6 = 3*4 = 2*2*3.

We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factored completely, 2*6 is not complete because 6 is not a prime number.

Exponents

To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.

Example B. Calculate.

32 =43 =

We write x*x*x…*x as xN where N is the number of x’s multiplied to itself. N is called the exponent, or the power of x.

Prime Factorization

To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as

12 = 2*6 = 3*4 = 2*2*3.

We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factored completely, 2*6 is not complete because 6 is not a prime number.

Exponents

To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.

Example B. Calculate.

32 = 3*3 = 943 =

We write x*x*x…*x as xN where N is the number of x’s multiplied to itself. N is called the exponent, or the power of x.

Prime Factorization

To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as

12 = 2*6 = 3*4 = 2*2*3.

We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factored completely, 2*6 is not complete because 6 is not a prime number.

Exponents

To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.

Example B. Calculate.

32 = 3*3 = 9 (note that 3*2 = 6)43 =

We write x*x*x…*x as xN where N is the number of x’s multiplied to itself. N is called the exponent, or the power of x.

Prime Factorization

To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as

12 = 2*6 = 3*4 = 2*2*3.

We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factored completely, 2*6 is not complete because 6 is not a prime number.

Exponents

To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.

Example B. Calculate.

32 = 3*3 = 9 (note that 3*2 = 6)43 = 4*4*4

We write x*x*x…*x as xN where N is the number of x’s multiplied to itself. N is called the exponent, or the power of x.

Prime Factorization

To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as

12 = 2*6 = 3*4 = 2*2*3.

We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factored completely, 2*6 is not complete because 6 is not a prime number.

Exponents

To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.

Example B. Calculate.

32 = 3*3 = 9 (note that 3*2 = 6)43 = 4*4*4 = 64

We write x*x*x…*x as xN where N is the number of x’s multiplied to itself. N is called the exponent, or the power of x.

Prime Factorization

To factor a number x means to write x as a product, i.e. write x as a*b for some a and b. For example 12 can be factored as

12 = 2*6 = 3*4 = 2*2*3.

We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factored completely, 2*6 is not complete because 6 is not a prime number.

Exponents

To simplify writing repetitive multiplication, we write 22 for 2*2, we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.

Example B. Calculate.

32 = 3*3 = 9 (note that 3*2 = 6)43 = 4*4*4 = 64 (note that 4*3 = 12)

We write x*x*x…*x as xN where N is the number of x’s multiplied to itself. N is called the exponent, or the power of x.

Prime Factorization

Numbers that are factored completely may be written using the exponential notation.

Prime Factorization

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 =

Prime Factorization

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3

Prime Factorization

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,

Prime Factorization

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 =

Prime Factorization

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25

Prime Factorization

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5

Prime Factorization

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

*52.

Prime Factorization

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

*52.

Prime Factorization

Each number has a unique form when its completely factored into prime factors and arranged in order.

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

*52.

Prime Factorization

Each number has a unique form when its completely factored into prime factors and arranged in order.

By “unique form” we mean that there is only one outcome.

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

*52.

Prime Factorization

Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

*52.

Prime Factorization

Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.

Larger numbers may be factored using a vertical format.

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

*52.

Prime Factorization

Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.

Larger numbers may be factored using a vertical format.

Example C. Factor 144 completely. (Vertical format)

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

*52.

Prime Factorization

Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.

Larger numbers may be factored using a vertical format.

Example C. Factor 144 completely. (Vertical format)

144

12 12

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

*52.

Prime Factorization

Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.

Larger numbers may be factored using a vertical format.

Example C. Factor 144 completely. (Vertical format)

144

12 12

3 4 3 4

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

*52.

Prime Factorization

Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.

Larger numbers may be factored using a vertical format.

Example C. Factor 144 completely. (Vertical format)

144

12 12

3 4 3 4

2 22 2

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

*52.

Prime Factorization

Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.

Larger numbers may be factored using a vertical format.

Example C. Factor 144 completely. (Vertical format)

144

12 12

3 4 3 4

2 22 2

Gather all the prime numbers at the end of the branches we have 144 = 3*3*2*2*2*2 = 3224.

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

*52.

Prime Factorization

Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.

Larger numbers may be factored using a vertical format.

Example C. Factor 144 completely. (Vertical format)

144

12 12

3 4 3 4

2 22 2

Gather all the prime numbers at the end of the branches we have 144 = 3*3*2*2*2*2 = 3224.

Note that we obtain the same answer regardless how we factor at each step.

Basic Laws

19. List the all the factors and the first 4 multiples of the following numbers. 6, 9, 10, 15, 16, 24, 30, 36, 42, 56, 60.

Calculate.1. 33 2. 42 3. 52 4. 53 5. 62 6. 63 7. 72

8. 82 9. 92 10. 102 11. 103 12. 104 13. 105

14. 1002 15. 1003 16. 1004 17. 112 18. 122

20. Factor completely and arrange the factors from smallest to the largest in the exponential notation: 4, 8, 12, 16, 18, 24, 27, 32, 36, 45, 48, 56, 60, 63, 72, 75, 81, 120.