Transcript of "123a-1-f1 prime numbers and factors"
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The Basics<br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />If x and y are two natural numbers and x can be divided by y, we say x is a multiple of y <br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />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 factorof x. <br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />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 factorof x. Hence 15 is a multiple of 3 (or 5), <br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />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 factorof x. Hence 15 is a multiple of 3 (or 5), and 3 (or 5) is a factor of 15. <br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />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 factorof 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.<br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />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 factorof 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.<br />A prime number is a natural number that can only be divided by 1 and itself. <br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />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 factorof 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.<br />A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. <br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />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 factorof 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.<br />A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. <br />Hence 2, 3, 5, 7, 11, 13,… are prime numbers. <br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />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 factorof 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.<br />A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. <br />Hence 2, 3, 5, 7, 11, 13,… are prime numbers. <br />15 is not a prime number because 15 can be divided by 3 or 5.<br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />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 factorof 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.<br />A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. <br />Hence 2, 3, 5, 7, 11, 13,… are prime numbers. <br />15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number. <br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />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 factorof 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.<br />A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. <br />Hence 2, 3, 5, 7, 11, 13,… are prime numbers. <br />15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number. <br />Example A. List the factors and the multiples of 12. <br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />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 factorof 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.<br />A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. <br />Hence 2, 3, 5, 7, 11, 13,… are prime numbers. <br />15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number. <br />Example A. List the factors and the multiples of 12. <br />The factors of 12:<br />The multiples of 12:<br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />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 factorof 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.<br />A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. <br />Hence 2, 3, 5, 7, 11, 13,… are prime numbers. <br />15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number. <br />Example A. List the factors and the multiples of 12. <br />The factors of 12: 1, 2, 3, 4, 6, and 12.<br />The multiples of 12:<br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />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 factorof 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.<br />A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. <br />Hence 2, 3, 5, 7, 11, 13,… are prime numbers. <br />15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number. <br />Example A. List the factors and the multiples of 12. <br />The factors of 12: 1, 2, 3, 4, 6, and 12.<br />The multiples of 12: 12=1x12, <br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />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 factorof 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.<br />A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. <br />Hence 2, 3, 5, 7, 11, 13,… are prime numbers. <br />15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number. <br />Example A. List the factors and the multiples of 12. <br />The factors of 12: 1, 2, 3, 4, 6, and 12.<br />The multiples of 12: 12=1x12, 24=2x12, <br />
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The Basics<br />Whole numbers 1, 2, 3, 4,.. are called natural numbers and they are used to track whole items. <br />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 factorof 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.<br />A prime number is a natural number that can only be divided by 1 and itself. Its factors consist of 1 and itself only. <br />Hence 2, 3, 5, 7, 11, 13,… are prime numbers. <br />15 is not a prime number because 15 can be divided by 3 or 5. The number 1 is not a prime number. <br />Example A. List the factors and the multiples of 12. <br />The factors of 12: 1, 2, 3, 4, 6, and 12.<br />The multiples of 12: 12=1x12, 24=2x12, 36=3x12, 48, 60, etc…<br />
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The Basics<br />To factor a number x means to write x as a product, i.e. write x as a*b for some a and b.<br />
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The Basics<br />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 <br />12 = 2*6 = 3*4 = 2*2*3. <br />
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The Basics<br />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 <br />12 = 2*6 = 3*4 = 2*2*3. <br />We say the factorization is complete if all the factors are prime numbers. <br />
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The Basics<br />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 <br />12 = 2*6 = 3*4 = 2*2*3. <br />We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, <br />
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The Basics<br />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 <br />12 = 2*6 = 3*4 = 2*2*3. <br />We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number.<br />
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The Basics<br />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 <br />12 = 2*6 = 3*4 = 2*2*3. <br />We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number.<br />Exponents<br />
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The Basics<br />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 <br />12 = 2*6 = 3*4 = 2*2*3. <br />We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number.<br />Exponents<br />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. <br />
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The Basics<br />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 <br />12 = 2*6 = 3*4 = 2*2*3. <br />We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number.<br />Exponents<br />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. <br />We write x*x*x…*x as xN where N is the number of x’s multiplied to itself. <br />
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The Basics<br />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 <br />12 = 2*6 = 3*4 = 2*2*3. <br />We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number.<br />Exponents<br />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. <br />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 powerof x.<br />
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The Basics<br />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 <br />12 = 2*6 = 3*4 = 2*2*3. <br />We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number.<br />Exponents<br />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. <br />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 powerof x.<br />Example B. Calculate.<br />32 =<br />43 =<br />
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The Basics<br />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 <br />12 = 2*6 = 3*4 = 2*2*3. <br />We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number.<br />Exponents<br />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. <br />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 powerof x.<br />Example B. Calculate.<br />32 = 3*3 = 9<br />43 =<br />
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The Basics<br />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 <br />12 = 2*6 = 3*4 = 2*2*3. <br />We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number.<br />Exponents<br />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. <br />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 powerof x.<br />Example B. Calculate.<br />32 = 3*3 = 9 (note that 3*2 = 6)<br />43 =<br />
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The Basics<br />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 <br />12 = 2*6 = 3*4 = 2*2*3. <br />We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number.<br />Exponents<br />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. <br />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 powerof x.<br />Example B. Calculate.<br />32 = 3*3 = 9 (note that 3*2 = 6)<br />43 = 4*4*4<br />
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The Basics<br />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 <br />12 = 2*6 = 3*4 = 2*2*3. <br />We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number.<br />Exponents<br />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. <br />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 powerof x.<br />Example B. Calculate.<br />32 = 3*3 = 9 (note that 3*2 = 6)<br />43 = 4*4*4 = 64<br />
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The Basics<br />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 <br />12 = 2*6 = 3*4 = 2*2*3. <br />We say the factorization is complete if all the factors are prime numbers. Hence 12=2*2*3 is factoredcompletely, 2*6 is not complete because 6 is not a prime number.<br />Exponents<br />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. <br />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 powerof x.<br />Example B. Calculate.<br />32 = 3*3 = 9 (note that 3*2 = 6)<br />43 = 4*4*4 = 64 (note that 4*3 = 12)<br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />Hence, factored completely, 12 =<br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />Hence, factored completely, 12 = 2*2*3<br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />Hence, factored completely, 12 = 2*2*3 = 22*3,<br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />Hence, factored completely, 12 = 2*2*3 = 22*3,<br />200 =<br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />Hence, factored completely, 12 = 2*2*3 = 22*3,<br />200 = 8*25<br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />Hence, factored completely, 12 = 2*2*3 = 22*3,<br />200 = 8*25 = 2*2*2*5*5<br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />Hence, factored completely, 12 = 2*2*3 = 22*3,<br />200 = 8*25 = 2*2*2*5*5 = 23*52.<br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />Hence, factored completely, 12 = 2*2*3 = 22*3,<br />200 = 8*25 = 2*2*2*5*5 = 23*52.<br />Each number has a unique form when its completely factored into prime factors and arranged in order. <br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />Hence, factored completely, 12 = 2*2*3 = 22*3,<br />200 = 8*25 = 2*2*2*5*5 = 23*52.<br />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. <br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />Hence, factored completely, 12 = 2*2*3 = 22*3,<br />200 = 8*25 = 2*2*2*5*5 = 23*52.<br />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. <br />Larger numbers may be factored using a vertical format. <br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />Hence, factored completely, 12 = 2*2*3 = 22*3,<br />200 = 8*25 = 2*2*2*5*5 = 23*52.<br />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. <br />Larger numbers may be factored using a vertical format. <br />Example C. Factor 144 completely. (Vertical format)<br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />Hence, factored completely, 12 = 2*2*3 = 22*3,<br />200 = 8*25 = 2*2*2*5*5 = 23*52.<br />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. <br />Larger numbers may be factored using a vertical format. <br />Example C. Factor 144 completely. (Vertical format)<br />144<br />12<br />12<br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />Hence, factored completely, 12 = 2*2*3 = 22*3,<br />200 = 8*25 = 2*2*2*5*5 = 23*52.<br />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. <br />Larger numbers may be factored using a vertical format. <br />Example C. Factor 144 completely. (Vertical format)<br />144<br />12<br />12<br />3<br />4<br />3<br />4<br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />Hence, factored completely, 12 = 2*2*3 = 22*3,<br />200 = 8*25 = 2*2*2*5*5 = 23*52.<br />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. <br />Larger numbers may be factored using a vertical format. <br />Example C. Factor 144 completely. (Vertical format)<br />144<br />12<br />12<br />3<br />4<br />3<br />4<br />2<br />2<br />2<br />2<br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />Hence, factored completely, 12 = 2*2*3 = 22*3,<br />200 = 8*25 = 2*2*2*5*5 = 23*52.<br />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. <br />Larger numbers may be factored using a vertical format. <br />Example C. Factor 144 completely. (Vertical format)<br />144<br />Gather all the prime numbers at the end of the branches we have 144 = 3*3*2*2*2*2 = 3224.<br />12<br />12<br />3<br />4<br />3<br />4<br />2<br />2<br />2<br />2<br />
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The Basics<br />Numbers that are factored completely may be written using the exponential notation.<br />Hence, factored completely, 12 = 2*2*3 = 22*3,<br />200 = 8*25 = 2*2*2*5*5 = 23*52.<br />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. <br />Larger numbers may be factored using a vertical format. <br />Example C. Factor 144 completely. (Vertical format)<br />144<br />Gather all the prime numbers at the end of the branches we have 144 = 3*3*2*2*2*2 = 3224.<br />12<br />12<br />3<br />4<br />3<br />4<br />Note that we obtain the same answer regardless how we factor at each step. <br />2<br />2<br />2<br />2<br />
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Basic Laws<br />Of the four arithmetic operations +, – , *, and ÷, <br />+, and * behave nicer than – or ÷. <br />
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Basic Laws<br />Of the four arithmetic operations +, – , *, and ÷, <br />+, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. <br />
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Basic Laws<br />Of the four arithmetic operations +, – , *, and ÷, <br />+, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. <br />
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Basic Laws<br />Of the four arithmetic operations +, – , *, and ÷, <br />+, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. <br />Associative Law forAddition and Multiplication<br />
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Basic Laws<br />Of the four arithmetic operations +, – , *, and ÷, <br />+, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. <br />Associative Law forAddition and Multiplication<br />( a + b ) + c = a + ( b + c ) <br />( a * b ) * c = a * ( b * c ) <br />Note: We are to perform the operations inside the “( )” first.<br />
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Basic Laws<br />Of the four arithmetic operations +, – , *, and ÷, <br />+, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. <br />Associative Law forAddition and Multiplication<br />( a + b ) + c = a + ( b + c ) <br />( a * b ) * c = a * ( b * c ) <br />Note: We are to perform the operations inside the “( )” first. <br />For example, (1 + 2) + 3<br />1 + (2 + 3)<br />
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Basic Laws<br />Of the four arithmetic operations +, – , *, and ÷, <br />+, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. <br />Associative Law forAddition and Multiplication<br />( a + b ) + c = a + ( b + c ) <br />( a * b ) * c = a * ( b * c ) <br />Note: We are to perform the operations inside the “( )” first. <br />For example, (1 + 2) + 3 = 3 + 3 = 6, <br />1 + (2 + 3)<br />
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Basic Laws<br />Of the four arithmetic operations +, – , *, and ÷, <br />+, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. <br />Associative Law forAddition and Multiplication<br />( a + b ) + c = a + ( b + c ) <br />( a * b ) * c = a * ( b * c ) <br />Note: We are to perform the operations inside the “( )” first. <br />For example, (1 + 2) + 3 = 3 + 3 = 6, <br />1 + (2 + 3) = 1 + 5 = 6. <br />
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Basic Laws<br />Of the four arithmetic operations +, – , *, and ÷, <br />+, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. <br />Associative Law forAddition and Multiplication<br />( a + b ) + c = a + ( b + c ) <br />( a * b ) * c = a * ( b * c ) <br />Note: We are to perform the operations inside the “( )” first. <br />For example, (1 + 2) + 3 = 3 + 3 = 6, which is the same as <br />1 + (2 + 3) = 1 + 5 = 6. <br />
60.
Basic Laws<br />Of the four arithmetic operations +, – , *, and ÷, <br />+, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. <br />Associative Law forAddition and Multiplication<br />( a + b ) + c = a + ( b + c ) <br />( a * b ) * c = a * ( b * c ) <br />Note: We are to perform the operations inside the “( )” first. <br />For example, (1 + 2) + 3 = 3 + 3 = 6, which is the same as <br />1 + (2 + 3) = 1 + 5 = 6. <br />Subtraction and division are not associative. <br />
61.
Basic Laws<br />Of the four arithmetic operations +, – , *, and ÷, <br />+, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. <br />Associative Law forAddition and Multiplication<br />( a + b ) + c = a + ( b + c ) <br />( a * b ) * c = a * ( b * c ) <br />Note: We are to perform the operations inside the “( )” first. <br />For example, (1 + 2) + 3 = 3 + 3 = 6, which is the same as <br />1 + (2 + 3) = 1 + 5 = 6. <br />Subtraction and division are not associative. <br />For example, ( 3 – 2 ) – 1<br /> 3 – ( 2 – 1 ) <br />
62.
Basic Laws<br />Of the four arithmetic operations +, – , *, and ÷, <br />+, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. <br />Associative Law forAddition and Multiplication<br />( a + b ) + c = a + ( b + c ) <br />( a * b ) * c = a * ( b * c ) <br />Note: We are to perform the operations inside the “( )” first. <br />For example, (1 + 2) + 3 = 3 + 3 = 6, which is the same as <br />1 + (2 + 3) = 1 + 5 = 6. <br />Subtraction and division are not associative. <br />For example, ( 3 – 2 ) – 1 = 0<br /> 3 – ( 2 – 1 )<br />
63.
Basic Laws<br />Of the four arithmetic operations +, – , *, and ÷, <br />+, and * behave nicer than – or ÷. We often take advantage of the this nice “behavior” of the addition and multiplication operations to get short cuts. However, be sure you don’t mistaken apply these short cuts to – and ÷. <br />Associative Law forAddition and Multiplication<br />( a + b ) + c = a + ( b + c ) <br />( a * b ) * c = a * ( b * c ) <br />Note: We are to perform the operations inside the “( )” first. <br />For example, (1 + 2) + 3 = 3 + 3 = 6, which is the same as <br />1 + (2 + 3) = 1 + 5 = 6. <br />Subtraction and division are not associative. <br />For example, ( 3 – 2 ) – 1 = 0 is different from <br /> 3 – ( 2 – 1 ) = 2. <br />
64.
Basic Laws<br />Commutative Law for Addition and Multiplication<br />
65.
Basic Laws<br />Commutative Law for Addition and Multiplication<br />a + b = b + a <br />a * b = b * a <br />
66.
Basic Laws<br />Commutative Law for Addition and Multiplication<br />a + b = b + a <br />a * b = b * a <br />For example, 3*4 = 4*3 = 12. <br />
67.
Basic Laws<br />Commutative Law for Addition and Multiplication<br />a + b = b + a <br />a * b = b * a <br />For example, 3*4 = 4*3 = 12. <br />Subtraction and division don't satisfy commutative laws. <br />
68.
Basic Laws<br />Commutative Law for Addition and Multiplication<br />a + b = b + a <br />a * b = b * a <br />For example, 3*4 = 4*3 = 12. <br />Subtraction and division don't satisfy commutative laws. <br />For example: 2 1 1 2.<br />
69.
Basic Laws<br />Commutative Law for Addition and Multiplication<br />a + b = b + a <br />a * b = b * a <br />For example, 3*4 = 4*3 = 12. <br />Subtraction and division don't satisfy commutative laws. <br />For example: 2 1 1 2.<br />From the above laws, we get the following important facts. <br />
70.
Basic Laws<br />Commutative Law for Addition and Multiplication<br />a + b = b + a <br />a * b = b * a <br />For example, 3*4 = 4*3 = 12. <br />Subtraction and division don't satisfy commutative laws. <br />For example: 2 1 1 2.<br />From the above laws, we get the following important facts. <br />When adding, the order of addition doesn’t matter.<br />
71.
Basic Laws<br />Commutative Law for Addition and Multiplication<br />a + b = b + a <br />a * b = b * a <br />For example, 3*4 = 4*3 = 12. <br />Subtraction and division don't satisfy commutative laws. <br />For example: 2 1 1 2.<br />From the above laws, we get the following important facts. <br />When adding, the order of addition doesn’t matter.<br />Hence to add a list numbers, it's easier to add the ones that add to multiples of 10 first.<br />
72.
Basic Laws<br />Commutative Law for Addition and Multiplication<br />a + b = b + a <br />a * b = b * a <br />For example, 3*4 = 4*3 = 12. <br />Subtraction and division don't satisfy commutative laws. <br />For example: 2 1 1 2.<br />From the above laws, we get the following important facts. <br />When adding, the order of addition doesn’t matter.<br />Hence to add a list numbers, it's easier to add the ones that add to multiples of 10 first.<br />Example D. <br />14 + 3 + 16 + 8 + 35 + 15 <br />
73.
Basic Laws<br />Commutative Law for Addition and Multiplication<br />a + b = b + a <br />a * b = b * a <br />For example, 3*4 = 4*3 = 12. <br />Subtraction and division don't satisfy commutative laws. <br />For example: 2 1 1 2.<br />From the above laws, we get the following important facts. <br />When adding, the order of addition doesn’t matter.<br />Hence to add a list numbers, it's easier to add the ones that add to multiples of 10 first.<br />Example D. <br />14 + 3 + 16 + 8 + 35 + 15 <br /> = 30 <br />
74.
Basic Laws<br />Commutative Law for Addition and Multiplication<br />a + b = b + a <br />a * b = b * a <br />For example, 3*4 = 4*3 = 12. <br />Subtraction and division don't satisfy commutative laws. <br />For example: 2 1 1 2.<br />From the above laws, we get the following important facts. <br />When adding, the order of addition doesn’t matter.<br />Hence to add a list numbers, it's easier to add the ones that add to multiples of 10 first.<br />Example D. <br />14 + 3 + 16 + 8 + 35 + 15 <br /> = 30 + 50 <br />
75.
Basic Laws<br />Commutative Law for Addition and Multiplication<br />a + b = b + a <br />a * b = b * a <br />For example, 3*4 = 4*3 = 12. <br />Subtraction and division don't satisfy commutative laws. <br />For example: 2 1 1 2.<br />From the above laws, we get the following important facts. <br />When adding, the order of addition doesn’t matter.<br />Hence to add a list numbers, it's easier to add the ones that add to multiples of 10 first.<br />Example D. <br />14 + 3 + 16 + 8 + 35 + 15 <br /> = 30 + 11 + 50 <br />
76.
Basic Laws<br />Commutative Law for Addition and Multiplication<br />a + b = b + a <br />a * b = b * a <br />For example, 3*4 = 4*3 = 12. <br />Subtraction and division don't satisfy commutative laws. <br />For example: 2 1 1 2.<br />From the above laws, we get the following important facts. <br />When adding, the order of addition doesn’t matter.<br />Hence to add a list numbers, it's easier to add the ones that add to multiples of 10 first.<br />ExampleD. <br />14 + 3 + 16 + 8 + 35 + 15 <br /> = 30 + 11 + 50 = 91 <br />
77.
Basic Laws<br />When multiplying, the order of multiplication doesn’t matter.<br />
78.
Basic Laws<br />When multiplying, the order of multiplication doesn’t matter.<br />When multiplying many numbers, always multiply them in pairs first. <br />
79.
Basic Laws<br />When multiplying, the order of multiplication doesn’t matter.<br />When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. <br />
80.
Basic Laws<br />When multiplying, the order of multiplication doesn’t matter.<br />When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. <br />Example E. <br />36<br />
81.
Basic Laws<br />When multiplying, the order of multiplication doesn’t matter.<br />When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. <br />Example E. <br />36 = 3*3*3*3*3*3<br /> = 9 * 9 * 9<br />
82.
Basic Laws<br />When multiplying, the order of multiplication doesn’t matter.<br />When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. <br />Example E. <br />36 = 3*3*3*3*3*3<br /> = 9 * 9 * 9<br /> = 81 * 9 = 729<br />
83.
Basic Laws<br />When multiplying, the order of multiplication doesn’t matter.<br />When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. <br />Example E. <br />36 = 3*3*3*3*3*3<br /> = 9 * 9 * 9<br /> = 81 * 9 = 729<br />Distributive Law :a*(b ± c) = a*b ± a*c<br />
84.
Basic Laws<br />When multiplying, the order of multiplication doesn’t matter.<br />When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. <br />Example E. <br />36 = 3*3*3*3*3*3<br /> = 9 * 9 * 9<br /> = 81 * 9 = 729<br />Distributive Law :a*(b ± c) = a*b ± a*c<br />Example F.<br />5*( 3 + 4 ) <br />
85.
Basic Laws<br />When multiplying, the order of multiplication doesn’t matter.<br />When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. <br />Example E. <br />36 = 3*3*3*3*3*3<br /> = 9 * 9 * 9<br /> = 81 * 9 = 729<br />Distributive Law :a*(b ± c) = a*b ± a*c<br />Example F.<br />5*( 3 + 4 ) = 5*7 = 35<br />
86.
Basic Laws<br />When multiplying, the order of multiplication doesn’t matter.<br />When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. <br />Example E. <br />36 = 3*3*3*3*3*3<br /> = 9 * 9 * 9<br /> = 81 * 9 = 729<br />Distributive Law :a*(b ± c) = a*b ± a*c<br />Example F.<br />5*( 3 + 4 ) = 5*7 = 35<br />or by distribute law, <br />5*( 3 + 4 ) <br />
87.
Basic Laws<br />When multiplying, the order of multiplication doesn’t matter.<br />When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. <br />Example E. <br />36 = 3*3*3*3*3*3<br /> = 9 * 9 * 9<br /> = 81 * 9 = 729<br />Distributive Law :a*(b ± c) = a*b ± a*c<br />Example F.<br />5*( 3 + 4 ) = 5*7 = 35<br />or by distribute law, <br />5*( 3 + 4 ) = 5*3 + 5*4 <br />
88.
Basic Laws<br />When multiplying, the order of multiplication doesn’t matter.<br />When multiplying many numbers, always multiply them in pairs first. This is useful for raising exponents. <br />Example E. <br />36 = 3*3*3*3*3*3<br /> = 9 * 9 * 9<br /> = 81 * 9 = 729<br />Distributive Law :a*(b ± c) = a*b ± a*c<br />Example F.<br />5*( 3 + 4 ) = 5*7 = 35<br />or by distribute law, <br />5*( 3 + 4 ) = 5*3 + 5*4 = 15 + 20 = 35<br />
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