Information about 2.4 prime numbers and factors w

Prime Numbers It we are to divide 6 pieces of candy into bag(s) so each bag has the same amount, we may do it in the following manner. Besides dividing them in the “obvious” ways: put all 6 candies into one bag, or put one candy into 6 separate bags, we may also divide the 6 pieces into smaller bags as: 3x2 or or 2 x 3 If we have 3 candies, we may bag them in the obvious manner but we can’t bag them into other smaller bags evenly. Because 3 can’t be divided into smaller groups, except as singles, we say that “3 is prime.” or On the other hand 6 = 2 x 3, i.e. 6 can be broken up into smaller groups, therefore 6 is not prime.

Prime Numbers Let’s define these important relations precisely.

Prime Numbers Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”.

Prime Numbers Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”. In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly.

Prime Numbers Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”. In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly. For example, 6 may be dividedly 1, 2, 3 and 6 evenly, hence 1, 2, 3, and 6 are factors of 6, or 6 is a multiple of 1, 2, 3 and 6.

Prime Numbers Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”. In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly. For example, 6 may be dividedly 1, 2, 3 and 6 evenly, hence 1, 2, 3, and 6 are factors of 6, or 6 is a multiple of 1, 2, 3 and 6. Note that the multiples of 6 are 6(= 1x6) , 12(= 2x6) , 18(= 3x6) etc.

Prime Numbers Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”. In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly. For example, 6 may be dividedly 1, 2, 3 and 6 evenly, hence 1, 2, 3, and 6 are factors of 6, or 6 is a multiple of 1, 2, 3 and 6. Note that the multiples of 6 are 6(= 1x6) , 12(= 2x6) , 18(= 3x6) etc. Example A. a. 12 pieces of candy are to be put into one or more bags evenly. How many piece(s) of candy in each bag are possible? (These numbers are the factors of 12.)

Prime Numbers Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”. In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly. For example, 6 may be dividedly 1, 2, 3 and 6 evenly, hence 1, 2, 3, and 6 are factors of 6, or 6 is a multiple of 1, 2, 3 and 6. Note that the multiples of 6 are 6(= 1x6) , 12(= 2x6) , 18(= 3x6) etc.. Example A. a. 12 pieces of candy are to be put into one or more bags evenly. How many piece(s) of candy in each bag are possible? (These numbers are the factors of 12.) We may have 1 piece/per bag,

Prime Numbers Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”. In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly. For example, 6 may be dividedly 1, 2, 3 and 6 evenly, hence 1, 2, 3, and 6 are factors of 6, or 6 is a multiple of 1, 2, 3 and 6. Note that the multiples of 6 are 6(= 1x6) , 12(= 2x6) , 18(= 3x6) etc. Example A. a. 12 pieces of candy are to be put into one or more bags evenly. How many piece(s) of candy in each bag are possible? (These numbers are the factors of 12.) We may have 1 piece/per bag, 2 pieces/bag,

Prime Numbers Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”. In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly. For example, 6 may be dividedly 1, 2, 3 and 6 evenly, hence 1, 2, 3, and 6 are factors of 6, or 6 is a multiple of 1, 2, 3 and 6. Note that the multiples of 6 are 6(= 1x6) , 12(= 2x6) , 18(= 3x6) etc. Example A. a. 12 pieces of candy are to be put into one or more bags evenly. How many piece(s) of candy in each bag are possible? (These numbers are the factors of 12.) We may have 1 piece/per bag, 2 pieces/bag, 3 pieces/ bag,

Prime Numbers Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”. In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly. For example, 6 may be dividedly 1, 2, 3 and 6 evenly, hence 1, 2, 3, and 6 are factors of 6, or 6 is a multiple of 1, 2, 3 and 6. Note that the multiples of 6 are 6(= 1x6) , 12(= 2x6) , 18(= 3x6) etc. Example A. a. 12 pieces of candy are to be put into one or more bags evenly. How many piece(s) of candy in each bag are possible? (These numbers are the factors of 12.) We may have 1 piece/per bag, 2 pieces/bag, 3 pieces/ bag, 4 pieces/bag, 6 pieces/bag, or 12 pieces/bag.

Prime Numbers Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”. In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly. For example, 6 may be dividedly 1, 2, 3 and 6 evenly, hence 1, 2, 3, and 6 are factors of 6, or 6 is a multiple of 1, 2, 3 and 6. Note that the multiples of 6 are 6(= 1x6) , 12(= 2x6) , 18(= 3x6) etc. Example A. a. 12 pieces of candy are to be put into one or more bags evenly. How many piece(s) of candy in each bag are possible? (These numbers are the factors of 12.) We may have 1 piece/per bag, 2 pieces/bag, 3 pieces/ bag, 4 pieces/bag, 6 pieces/bag, or 12 pieces/bag. So the factors of 12 are 1, 2, 3, 4, 6, and 12.

Prime Numbers b. A box contains bag(s) of candy and each bag has 12 pieces, how many candies in the box are possible? (These numbers are the multiples of 12.)

Prime Numbers b. A box contains bag(s) of candy and each bag has 12 pieces, how many candies in the box are possible? (These numbers are the multiples of 12.) There might be 1 bag or 2 bags or 3 bags, etc.. each with 12 pieces.

Prime Numbers b. A box contains bag(s) of candy and each bag has 12 pieces, how many candies in the box are possible? (These numbers are the multiples of 12.) There might be 1 bag or 2 bags or 3 bags, etc.. each with 12 pieces. So the box may have 12 =1x12, 24 =2x12, 36 =3x12, 48 =4x12, 60, ..., pieces, or that the multiples of 12 are 12, 24, 36, 48, …

Prime Numbers b. A box contains bag(s) of candy and each bag has 12 pieces, how many candies in the box are possible? (These numbers are the multiples of 12.) There might be 1 bag or 2 bags or 3 bags, etc.. each with 12 pieces. So the box may have 12 =1x12, 24 =2x12, 36 =3x12, 48 =4x12, 60, ..., pieces, or that the multiples of 12 are 12, 24, 36, 48, … A number C is a prime number if C only has 1 and itself as factors, i.e. C can be divided evenly only by 1 and itself.

Prime Numbers b. A box contains bag(s) of candy and each bag has 12 pieces, how many candies in the box are possible? (These numbers are the multiples of 12.) There might be 1 bag or 2 bags or 3 bags, etc.. each with 12 pieces. So the box may have 12 =1x12, 24 =2x12, 36 =3x12, 48 =4x12, 60, ..., pieces, or that the multiples of 12 are 12, 24, 36, 48, … A number C is a prime number if C only has 1 and itself as factors, i.e. C can be divided evenly only by 1 and itself. The following are the first few prime numbers: 2, 3, 5, 7, 11,13,..;

Prime Numbers b. A box contains bag(s) of candy and each bag has 12 pieces, how many candies in the box are possible? (These numbers are the multiples of 12.) There might be 1 bag or 2 bags or 3 bags, etc.. each with 12 pieces. So the box may have 12 =1x12, 24 =2x12, 36 =3x12, 48 =4x12, 60, ..., pieces, or that the multiples of 12 are 12, 24, 36, 48, … A number C is a prime number if C only has 1 and itself as factors, i.e. C can be divided evenly only by 1 and itself. The following are the first few prime numbers: 2, 3, 5, 7, 11,13,..; but 4 = 2 x 2, 6 = 2 x 3, 8 = 2 x 2 x 2, 9 = 3 x 3, are not prime.

Prime Numbers b. A box contains bag(s) of candy and each bag has 12 pieces, how many candies in the box are possible? (These numbers are the multiples of 12.) There might be 1 bag or 2 bags or 3 bags, etc.. each with 12 pieces. So the box may have 12 =1x12, 24 =2x12, 36 =3x12, 48 =4x12, 60, ..., pieces, or that the multiples of 12 are 12, 24, 36, 48, … A number C is a prime number if C only has 1 and itself as factors, i.e. C can be divided evenly only by 1 and itself. The following are the first few prime numbers: 2, 3, 5, 7, 11,13,..; but 4 = 2 x 2, 6 = 2 x 3, 8 = 2 x 2 x 2, 9 = 3 x 3, are not prime. The problem of determining easily which numbers are prime and which are not, is a major problem in mathematics.

Prime Numbers b. A box contains bag(s) of candy and each bag has 12 pieces, how many candies in the box are possible? (These numbers are the multiples of 12.) There might be 1 bag or 2 bags or 3 bags, etc.. each with 12 pieces. So the box may have 12 =1x12, 24 =2x12, 36 =3x12, 48 =4x12, 60, ..., pieces, or that the multiples of 12 are 12, 24, 36, 48, … A number C is a prime number if C only has 1 and itself as factors, i.e. C can be divided evenly only by 1 and itself. The following are the first few prime numbers: 2, 3, 5, 7, 11,13,..; but 4 = 2 x 2, 6 = 2 x 3, 8 = 2 x 2 x 2, 9 = 3 x 3, are not prime. The problem of determining easily which numbers are prime and which are not, is a major problem in mathematics. The link below has a good animation of one method of trying to determine which numbers are prime and which are not. http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

Exponents and Prime Factoring Below we will write “A x B” as “A*B” for shorter notation.

Exponents and Prime Factoring Below we will write “A x B” as “A*B” for shorter notation. Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on.

Exponents and Prime Factoring Below we will write “A x B” as “A*B” for shorter notation. Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on. We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself.

Exponents and Prime Factoring Below we will write “A x B” as “A*B” for shorter notation. Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on. We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself. N is called the exponent, or the power and x is called the base.

Exponents and Prime Factoring Below we will write “A x B” as “A*B” for shorter notation. Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on. We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself. N is called the exponent, or the power and x is called the base. Example B. Calculate. 32 = 43 =

Exponents and Prime Factoring Below we will write “A x B” as “A*B” for shorter notation. Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on. We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself. N is called the exponent, or the power and x is called the base. Example B. Calculate. 32 = 3*3 = 9 43 =

Exponents and Prime Factoring Below we will write “A x B” as “A*B” for shorter notation. Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on. We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself. N is called the exponent, or the power and x is called the base. Example B. Calculate. (note that 32 is not 3*2 = 6) 32 = 3*3 = 9 43 =

Exponents and Prime Factoring Below we will write “A x B” as “A*B” for shorter notation. Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on. We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself. N is called the exponent, or the power and x is called the base. Example B. Calculate. (note that 32 is not 3*2 = 6) 32 = 3*3 = 9 43 = 4*4*4 = 64

Exponents and Prime Factoring Below we will write “A x B” as “A*B” for shorter notation. Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on. We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself. N is called the exponent, or the power and x is called the base. Example B. Calculate. (note that 32 is not 3*2 = 6) 32 = 3*3 = 9 43 = 4*4*4 = 64 (note that 43 is not 4*3 = 12)

Exponents and Prime Factoring Below we will write “A x B” as “A*B” for shorter notation. Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on. We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself. N is called the exponent, or the power and x is called the base. Example B. Calculate. (note that 32 is not 3*2 = 6) 32 = 3*3 = 9 43 = 4*4*4 = 64 (note that 43 is not 4*3 = 12) The phrase to factor a number C means to write C as a product, i.e. to write C as # * # *** # where # might be any number.

Exponents and Prime Factoring Below we will write “A x B” as “A*B” for shorter notation. Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on. We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself. N is called the exponent, or the power and x is called the base. Example B. Calculate. (note that 32 is not 3*2 = 6) 32 = 3*3 = 9 43 = 4*4*4 = 64 (note that 43 is not 4*3 = 12) The phrase to factor a number C means to write C as a product, i.e. to write C as # * # *** # where # might be any number. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3.

Exponents and Prime Factoring Below we will write “A x B” as “A*B” for shorter notation. Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on. We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself. N is called the exponent, or the power and x is called the base. Example B. Calculate. (note that 32 is not 3*2 = 6) 32 = 3*3 = 9 43 = 4*4*4 = 64 (note that 43 is not 4*3 = 12) The phrase to factor a number C means to write C as a product, i.e. to write C as # * # *** # where # might be any number. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3. A factorization is complete if all the factors that appear in the multiplication are prime numbers.

Exponents and Prime Factoring Below we will write “A x B” as “A*B” for shorter notation. Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on. We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself. N is called the exponent, or the power and x is called the base. Example B. Calculate. (note that 32 is not 3*2 = 6) 32 = 3*3 = 9 43 = 4*4*4 = 64 (note that 43 is not 4*3 = 12) The phrase to factor a number C means to write C as a product, i.e. to write C as # * # *** # where # might be any number. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3. A factorization is complete if all the factors that appear in the multiplication are prime numbers. So 12 = 2*2*3 is factored completely because 2 and 3 are prime,

Exponents and Prime Factoring Below we will write “A x B” as “A*B” for shorter notation. Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on. We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself. N is called the exponent, or the power and x is called the base. Example B. Calculate. (note that 32 is not 3*2 = 6) 32 = 3*3 = 9 43 = 4*4*4 = 64 (note that 43 is not 4*3 = 12) The phrase to factor a number C means to write C as a product, i.e. to write C as # * # *** # where # might be any number. For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3. A factorization is complete if all the factors that appear in the multiplication are prime numbers. So 12 = 2*2*3 is factored completely because 2 and 3 are prime, but 12 = 2* 6 is not factored completely because 6 is not a prime.

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

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

Exponents 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

Exponents 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.

Exponents 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.

Exponents 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.

Exponents 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.

Exponents 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)

Exponents 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) 144 12 12

Exponents 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) 144 12 3 4 3 12 4

Exponents 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) 144 12 3 12 4 3 2 2 2 4 2

Exponents 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) 144 Gather all the prime numbers at the end of the branches we have 12 12 144 = 3*3*2*2*2*2 = 32 * 24. 3 4 3 4 2 2 2 2

Exponents 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) 144 Gather all the prime numbers at the end of the branches we have 12 12 144 = 3*3*2*2*2*2 = 32 * 24. 3 4 3 4 Note that we obtain the same answer regardless how we factor at each step. 2 2 2 2

Basic Laws We summarize the basic laws of + and * operations again.

Basic Laws We summarize the basic laws of + and * operations again. The Commutative Law for Addition and Multiplication The Associative Law for Addition and Multiplication

Basic Laws We summarize the basic laws of + and * operations again. The Commutative Law for Addition and Multiplication a+b=b+a a*b=b*a The Associative Law for Addition and Multiplication

Basic Laws We summarize the basic laws of + and * operations again. The Commutative Law for Addition and Multiplication a+b=b+a a*b=b*a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12. The Associative Law for Addition and Multiplication

Basic Laws We summarize the basic laws of + and * operations again. The Commutative Law for Addition and Multiplication a+b=b+a a*b=b*a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12. The Associative Law for Addition and Multiplication (a+b) +c=a+(b+c) (a*b)*c = a*(b*c)

Basic Laws We summarize the basic laws of + and * operations again. The Commutative Law for Addition and Multiplication a+b=b+a a*b=b*a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12. The Associative Law for Addition and Multiplication (a+b) +c=a+(b+c) (a*b)*c = a*(b*c) For example, (1 + 2) + 3 = 3 + 3 = 6, is the same as 1 + (2 + 3) = 1 + 5 = 6.

Basic Laws We summarize the basic laws of + and * operations again. The Commutative Law for Addition and Multiplication a+b=b+a a*b=b*a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12. The Associative Law for Addition and Multiplication (a+b) +c=a+(b+c) (a*b)*c = a*(b*c) For example, (1 + 2) + 3 = 3 + 3 = 6, is the same as 1 + (2 + 3) = 1 + 5 = 6. These laws allow us to + or * numbers in any order we wish.

Basic Laws We summarize the basic laws of + and * operations again. The Commutative Law for Addition and Multiplication a+b=b+a a*b=b*a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12. The Associative Law for Addition and Multiplication (a+b) +c=a+(b+c) (a*b)*c = a*(b*c) For example, (1 + 2) + 3 = 3 + 3 = 6, is the same as 1 + (2 + 3) = 1 + 5 = 6. These laws allow us to + or * numbers in any order we wish. Example D. Calculate. 14 + 3 + 16 + 8 + 35 + 15

Basic Laws We summarize the basic laws of + and * operations again. The Commutative Law for Addition and Multiplication a+b=b+a a*b=b*a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12. The Associative Law for Addition and Multiplication (a+b) +c=a+(b+c) (a*b)*c = a*(b*c) For example, (1 + 2) + 3 = 3 + 3 = 6, is the same as 1 + (2 + 3) = 1 + 5 = 6. These laws allow us to + or * numbers in any order we wish. Example D. Calculate. 14 + 3 + 16 + 8 + 35 + 15 = 30

Basic Laws We summarize the basic laws of + and * operations again. The Commutative Law for Addition and Multiplication a+b=b+a a*b=b*a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12. The Associative Law for Addition and Multiplication (a+b) +c=a+(b+c) (a*b)*c = a*(b*c) For example, (1 + 2) + 3 = 3 + 3 = 6, is the same as 1 + (2 + 3) = 1 + 5 = 6. These laws allow us to + or * numbers in any order we wish. Example D. Calculate. 14 + 3 + 16 + 8 + 35 + 15 = 30 + 50

Basic Laws We summarize the basic laws of + and * operations again. The Commutative Law for Addition and Multiplication a+b=b+a a*b=b*a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12. The Associative Law for Addition and Multiplication (a+b) +c=a+(b+c) (a*b)*c = a*(b*c) For example, (1 + 2) + 3 = 3 + 3 = 6, is the same as 1 + (2 + 3) = 1 + 5 = 6. These laws allow us to + or * numbers in any order we wish. Example D. Calculate. 14 + 3 + 16 + 8 + 35 + 15 = 30 + 11 + 50

Basic Laws We summarize the basic laws of + and * operations again. The Commutative Law for Addition and Multiplication a+b=b+a a*b=b*a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12. The Associative Law for Addition and Multiplication (a+b) +c=a+(b+c) (a*b)*c = a*(b*c) For example, (1 + 2) + 3 = 3 + 3 = 6, is the same as 1 + (2 + 3) = 1 + 5 = 6. These laws allow us to + or * numbers in any order we wish. Example D. Calculate. 14 + 3 + 16 + 8 + 35 + 15 = 30 + 11 + 50 = 91

Basic Laws However, subtraction and division are not commutative nor associative.

Basic Laws However, subtraction and division are not commutative nor associative. For example, 2 1 1 2,

Basic Laws However, subtraction and division are not commutative nor associative. For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2.

Basic Laws However, subtraction and division are not commutative nor associative. For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back.

Basic Laws However, subtraction and division are not commutative nor associative. For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back. Distributive Law: a*(b . c) = a*b a*c

Basic Laws However, subtraction and division are not commutative nor associative. For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back. Distributive Law: a*(b c) = a*b a*c Example E. Do 5*( 3 + 4 ) two different ways and show the . results are the same.

Basic Laws However, subtraction and division are not commutative nor associative. For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back. Distributive Law: a*(b c) = a*b a*c Example E. Do 5*( 3 + 4 ) two different ways and show the . results are the same. Do the ( ) first:

Basic Laws However, subtraction and division are not commutative nor associative. For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back. Distributive Law: a*(b c) = a*b a*c Example E. Do 5*( 3 + 4 ) two different ways and show the . results are the same. Do the ( ) first: 5*( 3 + 4 ) = 5*7 = 35,

Basic Laws However, subtraction and division are not commutative nor associative. For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back. Distributive Law: a*(b c) = a*b a*c Example E. Do 5*( 3 + 4 ) two different ways and show the . results are the same. Do the ( ) first: 5*( 3 + 4 ) = 5*7 = 35, Distribute the 5 first:

Basic Laws However, subtraction and division are not commutative nor associative. For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back. Distributive Law: a*(b c) = a*b a*c Example E. Do 5*( 3 + 4 ) two different ways and show the . results are the same. Do the ( ) first: 5*( 3 + 4 ) = 5*7 = 35, Distribute the 5 first: 5*( 3 + 4 ) = 5*3 + 5*4

Basic Laws However, subtraction and division are not commutative nor associative. For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back. Distributive Law: a*(b c) = a*b a*c Example E. Do 5*( 3 + 4 ) two different ways and show the . results are the same. Do the ( ) first: 5*( 3 + 4 ) = 5*7 = 35, Distribute the 5 first: 5*( 3 + 4 ) = 5*3 + 5*4 = 15 + 20 = 35

Basic Laws However, subtraction and division are not commutative nor associative. For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back. Distributive Law: a*(b c) = a*b a*c Example E. Do 5*( 3 + 4 ) two different ways and show the . results are the same. Do the ( ) first: 5*( 3 + 4 ) = 5*7 = 35, Distribute the 5 first: 5*( 3 + 4 ) = 5*3 + 5*4 = 15 + 20 = 35 the outcomes are the same as stated in the distributive law.

B. 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 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. 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. C. Multiply in two ways to find the correct answer. 21. 3 * 5 * 4 * 2 23. 6 * 15 * 3 * 2 22. 6 * 5 * 4 * 3 24. 27. 29. 31. 36. 7*5*5*4 25. 6 * 7 * 4 * 3 2 * 25 * 3 * 4 * 2 28. 3 * 2 3*5*2*5*2*4 30. 4 * 2 24 32. 25 33. 26 37. 210 29 38. 34 26. 9 * 3 * 4 * 4 * 3 * 3 * 2 * 4 * 3 * 15 * 8 * 4© 34. 27 35. 28 39. 35 40. 36 © F. Ma

Exercise A. Do the following problems two ways. * Add the following by summing the multiples of 10 first. * Add by adding in the order. to find the correct answer. 1. 3 + 5 + 7 2. 8 + 6 + 2 3. 1 + 8 + 9 4. 3 + 5 + 15 5. 9 + 14 + 6 6. 22 + 5 + 8 7. 16 + 5 + 4 + 3 8. 4 + 13 + 5 + 7 9. 19 + 7 + 1 + 3 10. 4 + 5 + 17 + 3 11. 23 + 5 + 17 + 3 12. 22 + 5 + 13 + 28 13. 35 + 6 + 15 + 7 + 14 14. 42 + 5 + 18 + 12 15. 21 + 16 + 19 + 7 + 44 16. 53 + 5 + 18 + 27 + 22 17. 155 + 16 + 25 + 7 + 344 18. 428 + 3 + 32 + 227 + 22 Hence to add a list numbers, it's easier to add the ones that add to multiples of 10 first. Example D.

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