Prime Factorization

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

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.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.

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.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.

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.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.

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.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)

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.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)

14412 12

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.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)

14412 12

3 4 3 4

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.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)

14412 12

3 4 3 42 22 2

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.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)

14412 12

3 4 3 42 22 2

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

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. 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)

14412 12

3 4 3 42 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.

Definition of LCMThe least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers.

LCM

Definition of LCMThe least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example D. Find the LCM of 4 and 6.

LCM

Definition of LCMThe least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example D. Find the LCM of 4 and 6.The multiples of 4 are 4, 8, 12, 16, 20, 24, …

LCM

Definition of LCMThe least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example D. Find the LCM of 4 and 6.The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,…

LCM

Definition of LCMThe least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example D. Find the LCM of 4 and 6.The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12,

LCM

Definition of LCMThe least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example D. Find the LCM of 4 and 6.The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, so LCM{4, 6 } = 12.

LCM

Definition of LCMThe least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example D. Find the LCM of 4 and 6.The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, so LCM{4, 6 } = 12. We may improve the above listing-method for finding the LCM.

LCM

Definition of LCMThe least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example D. Find the LCM of 4 and 6.The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, so LCM{4, 6 } = 12. We may improve the above listing-method for finding the LCM. Given two or more numbers, we start with the largest number,

LCM

Definition of LCMThe least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example D. Find the LCM of 4 and 6.The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, so LCM{4, 6 } = 12. We may improve the above listing-method for finding the LCM. Given two or more numbers, we start with the largest number, list its multiples in order until we find the one that can divide all the other numbers.

LCM

Definition of LCMThe least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example D. Find the LCM of 4 and 6.The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, so LCM{4, 6 } = 12.

Example E. Find the LCM of 8, 9, and 12.

We may improve the above listing-method for finding the LCM. Given two or more numbers, we start with the largest number, list its multiples in order until we find the one that can divide all the other numbers.

LCM

Definition of LCMThe least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example D. Find the LCM of 4 and 6.The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, so LCM{4, 6 } = 12.

Example E. Find the LCM of 8, 9, and 12. The largest number is 12 and the multiples of 12 are 12, 24, 36, 48, 60, 72, 84 …

We may improve the above listing-method for finding the LCM. Given two or more numbers, we start with the largest number, list its multiples in order until we find the one that can divide all the other numbers.

LCM

Definition of LCMThe least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example D. Find the LCM of 4 and 6.The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, so LCM{4, 6 } = 12.

Example E. Find the LCM of 8, 9, and 12. The largest number is 12 and the multiples of 12 are 12, 24, 36, 48, 60, 72, 84 … The first number that is also a multiple of 8 and 9 is 72 so LCM{8, 9, 12} = 72.

We may improve the above listing-method for finding the LCM. Given two or more numbers, we start with the largest number, list its multiples in order until we find the one that can divide all the other numbers.

LCM

But when the LCM is large, the listing method is cumbersome. LCM

But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead.

LCM

Example F. Construct the LCM of {8, 15, 18}.

But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead.

LCM

To construct the LCM: a. Factor each number completely

Example F. Construct the LCM of {8, 15, 18}.

But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead.

LCM

To construct the LCM: a. Factor each number completely

Example F. Construct the LCM of {8, 15, 18}.Factor each number completely,

But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead.

LCM

To construct the LCM: a. Factor each number completely

Example F. Construct the LCM of {8, 15, 18}.Factor each number completely, 8 = 23

But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead.

LCM

To construct the LCM: a. Factor each number completely

Example F. Construct the LCM of {8, 15, 18}.Factor each number completely, 8 = 23

15 = 3 * 5

But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead.

LCM

To construct the LCM: a. Factor each number completely

Example F. Construct the LCM of {8, 15, 18}.Factor each number completely, 8 = 23

15 = 3 * 518 = 2 * 32

But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead.

LCM

To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in

the factorizations. The LCM is their product.Example F. Construct the LCM of {8, 15, 18}.Factor each number completely, 8 = 23

15 = 3 * 518 = 2 * 32

But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead.

LCM

To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in

the factorizations. The LCM is their product.

But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead.

Example F. Construct the LCM of {8, 15, 18}.Factor each number completely, 8 = 23

15 = 3 * 518 = 2 * 32 The prime factors appeared in the factorizations are 2, 3 and 5.

LCM

To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in

the factorizations. The LCM is their product.

But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead.

Example F. Construct the LCM of {8, 15, 18}.Factor each number completely, 8 = 23

15 = 3 * 518 = 2 * 32 The prime factors appeared in the factorizations are 2, 3 and 5. The highest degree of each prime factor are: 23,

LCM

To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in

the factorizations. The LCM is their product.

But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead.

Example F. Construct the LCM of {8, 15, 18}.Factor each number completely, 8 = 23

15 = 3 * 518 = 2 * 32 The prime factors appeared in the factorizations are 2, 3 and 5. The highest degree of each prime factor are: 23, 32

LCM

To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in

the factorizations. The LCM is their product.

But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead.

Example F. Construct the LCM of {8, 15, 18}.Factor each number completely, 8 = 23

15 = 3 * 518 = 2 * 32 The prime factors appeared in the factorizations are 2, 3 and 5. The highest degree of each prime factor are: 23, 32 and 5,

LCM

To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in

the factorizations. The LCM is their product.

But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead.

Example F. Construct the LCM of {8, 15, 18}.Factor each number completely, 8 = 23

15 = 3 * 518 = 2 * 32 The prime factors appeared in the factorizations are 2, 3 and 5. The highest degree of each prime factor are: 23, 32 and 5, so LCM{8, 15, 18} = 23*32*5

LCM

To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in

the factorizations. The LCM is their product.

But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead.

LCM

Example F. Construct the LCM of {8, 15, 18}.Factor each number completely, 8 = 23

15 = 3 * 518 = 2 * 32 The prime factors appeared in the factorizations are 2, 3 and 5. The highest degree of each prime factor are: 23, 32 and 5, so LCM{8, 15, 18} = 23*32*5 = 8*9*5 = 360.

The Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list.

Methods of Finding LCM

The Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list. The following is an example that illustrates the principle behind this method.

Methods of Finding LCM

Methods of Finding LCMThe Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list. The following is an example that illustrates the principle behind this method. Example C. A student wants to take enough courses so she meets the requirement to apply to colleges A, B, and C. The following is a table of requirements for each college, how many years of each subject does she need?

Science Math. English Arts

A 2 yrs 3 yrs 3 yrs

B 3 yrs 3 yrs 2 yrs

C 2 yrs 2 yrs 4 yrs 1 yrs

Methods of Finding LCMThe Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list. The following is an example that illustrates the principle behind this method. Example C. A student wants to take enough courses so she meets the requirement to apply to colleges A, B, and C. The following is a table of requirements for each college, how many years of each subject does she need?

Science Math. English Arts

A 2 yrs 3 yrs 3 yrs

B 3 yrs 3 yrs 2 yrs

C 2 yrs 2 yrs 4 yrs 1 yrs

Methods of Finding LCMThe Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list. The following is an example that illustrates the principle behind this method. Example C. A student wants to take enough courses so she meets the requirement to apply to colleges A, B, and C. The following is a table of requirements for each college, how many years of each subject does she need?

Science Math. English Arts

A 2 yrs 3 yrs 3 yrs

B 3 yrs 3 yrs 2 yrs

C 2 yrs 2 yrs 4 yrs 1 yrs

So the student needs to take:

Methods of Finding LCMThe Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list. The following is an example that illustrates the principle behind this method. Example C. A student wants to take enough courses so she meets the requirement to apply to colleges A, B, and C. The following is a table of requirements for each college, how many years of each subject does she need?

Science-3 yr Math-3yrs English-4 yrs Arts-1yr

Factorization

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.

LCMFind the LCM.1. a.{6, 8} b. {6, 9} c. {3, 4} d. {4, 10} 2. a.{5, 6, 8} b. {4, 6, 9} c. {3, 4, 5} d. {4, 6, 10} 3. a.{6, 8, 9} b. {6, 9, 10} c. {4, 9, 10}

d. {6, 8, 10} 4. a.{4, 8, 15} b. {8, 9, 12} c. {6, 9, 15} 5. a.{6, 8, 15} b. {8, 9, 15} c. {6, 9, 16} 6. a.{8, 12, 15} b. { 9, 12, 15} c. { 9, 12, 16} 7. a.{8, 12, 18} b. {8, 12, 20} c. { 12, 15, 16} 8. a.{8, 12, 15, 18} b. {8, 12, 16, 20} 9. a.{8, 15, 18, 20} b. {9, 16, 20, 24}