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Vocabulary for Unit 1 Vocabulary for Unit 1 Prime Number – A number greater than 1 that has only 2 factors, 1 and itself. Composite Number – A number that has 3 or more factors. Factor – A number that divides evenly into another number without a remainder. Greatest Common Factor (GCF) – The largest number that is a factor for 2 or more numbers. Multiple – The product of a number and any other whole number. Least Common Multiple (LCM) – The smallest number that is a multiple of 2 or more numbers. Factor Tree – A diagram used to decompose a composite number into its prime factorization. Prime Factorization – An expression of a whole number as a product of prime numbers. Exponent – The number of times a base number is multiplied by itself. Prime Number – A number greater than 1 that has only 2 factors, 1 and itself. Composite Number – A number that has 3 or more factors. Factor – A number that divides evenly into another number without a remainder. Greatest Common Factor (GCF) – The largest number that is a factor for 2 or more numbers. Multiple – The product of a number and any other whole number. Least Common Multiple (LCM) – The smallest number that is a multiple of 2 or more numbers. Factor Tree – A diagram used to decompose a composite number into its prime factorization. Prime Factorization – An expression of a whole number as a product of prime numbers. Exponent – The number of times a base number is multiplied by itself. 3 4 16 17 29 30 42 43 7 20 33 46 8 21 34 47 9 22 35 48 10 23 36 49 11 24 37 50 Prime and Composite Numbers 5 6 18 19 31 32 44 45 12 25 38 51 13 26 39 52 14 27 40 53 2 15 28 41 3 4 16 17 29 30 42 43 5 6 18 19 31 32 44 45 7 20 33 46 8 21 34 47 9 22 35 48 10 23 36 49 11 24 37 50 12 25 38 51 13 26 39 52 14 27 40 53 Then go to the next remaining number after 2. Circle it and cross out all its multiples other than itself. Repeat until every number is either circled or crossed out. Step 2 Circle 2 and cross out all multiples of 2 other than the number 2. Step 1 2 15 28 41 Then go to the next remaining number after 2. Circle it and cross out all its multiples other than itself. Repeat until every number is either circled or crossed out. Step 2 Circle 2 and cross out all multiples of 2 other than the number 2. Step 1 Prime and Composite Numbers These are prime numbers. 47 23 7 31 These are prime numbers. 101 59 13 29 67 5 11 3 23 These are composite numbers. 87 33 14 27 16 9 25 15 2 4 19 110 6 1 25 18 61 What is a prime number? What is a composite number? 31 57 111 87 33 Circle the prime numbers. Underline the composite numbers. 2 7 101 59 13 29 67 5 11 3 These are composite numbers. 51 22 47 27 51 16 22 9 25 15 2 4 Circle the prime numbers. Underline the composite numbers. 0 90 14 2 19 110 6 1 25 18 61 What is a prime number? What is a composite number? 57 111 0 90 Divisibility Rules 2: If the last digit is even (0, 2, 4, 6, or 8). 24 – yes 3: 30 – yes Divisibility Rules 2: 67 - no If the sum of the digits is divisible by 3. 24 – yes 3: 57: 5 + 7 = 12 & 12 is divisible by 3 4: If the last 2 digits are divisible by 4. 132: 32 ÷ 4 = 8 5: 85 – yes 6: 120 – yes 4: 5: If the sum of the digits is divisible by 9. 585: 5 + 8 + 5 = 18 & 18 is divisible by 9 10: If the last digit is 0. 90 – yes 680 – yes If the last 2 digits are divisible by 4. 6: X 226: 26 ÷ 4 = 6.5 If the last digit is 0 or 5. 85 – yes 54: 5 + 4 = 9 & 9 is divisible by 3 9: 67 - no If the sum of the digits is divisible by 3. 132: 32 ÷ 4 = 8 53 - no If the number is divisible by 2 AND 3. 30 – yes 57: 5 + 7 = 12 & 12 is divisible by 3 X 226: 26 ÷ 4 = 6.5 If the last digit is 0 or 5. If the last digit is even (0, 2, 4, 6, or 8). 120 – yes 53 - no If the number is divisible by 2 AND 3. 54: 5 + 4 = 9 & 9 is divisible by 3 9: If the sum of the digits is divisible by 9. 585: 5 + 8 + 5 = 18 & 18 is divisible by 9 10: If the last digit is 0. 104- no 90 – yes 680 – yes 104- no Factor Notes Factor Notes 1 is a factor of every number. 1 is a factor of every number. 0 is NEVER a factor of any number because 0 is NEVER a factor of any number because numbers cannot be divided by 0. numbers cannot be divided by 0. Examples: Examples: Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40 Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 When would I need to use factors? When would I need to use factors? A large group of people want to split up to make A large group of people want to split up to make smaller groups to walk around an amusement park. smaller groups to walk around an amusement park. What are the possibilities if the group is 18? What are the possibilities if the group is 18? Greatest Common Factor (GCF) – The largest Greatest Common Factor (GCF) – The largest number that is a factor for 2 or more numbers. number that is a factor for 2 or more numbers. What is the GCF of 16 and 24? What is the GCF of 16 and 24? 16: 1, 2, 4, 8, 16 16: 1, 2, 4, 8, 16 24: 1, 2, 3, 4, 6, 8, 12, 24 24: 1, 2, 3, 4, 6, 8, 12, 24 Common Factors are 1, 2, 4, 8 Common Factors are 1, 2, 4, 8 8 is the GCF 8 is the GCF What is the GCF of 20 and 40? What is the GCF of 20 and 40? 20: 1, 2, 4, 5, 10, 20 20: 1, 2, 4, 5, 10, 20 40: 1, 2, 4, 5, 8, 10, 20, 40 40: 1, 2, 4, 5, 8, 10, 20, 40 Common Factors are 1, 2, 4, 5, 10, 20 Common Factors are 1, 2, 4, 5, 10, 20 20 is the GCF 20 is the GCF If the GCF is 1, then the numbers are relatively prime. If the GCF is 1, then the numbers are relatively prime. Ex. 15: 1, 3, 5, 15 8: 1, 2, 4, 8 Ex. 15: 1, 3, 5, 15 8: 1, 2, 4, 8 Multiple – The product of a number and any other whole Multiple – The product of a number and any other whole number. number. Multiples of 6 are 6, 12, 18, 24, 30 and so on…. Multiples of 6 are 6, 12, 18, 24, 30 and so on…. How can I check to see if a number is a multiple of another How can I check to see if a number is a multiple of another number? Divide! No remainder = Multiple number? Divide! No remainder = Multiple 24 24 Ex. 6 144 Ex. 6 144 -12 So, 144 is a multiple of 6. -12 So, 144 is a multiple of 6. 24 24 -24 -24 0 0 Least Common Multiple (LCM) – The smallest number Least Common Multiple (LCM) – The smallest number that is a multiple of 2 or more numbers. that is a multiple of 2 or more numbers. What is the LCM of 8 and 12? What is the LCM of 8 and 12? Multiples of 8: 8, 16, 24, 32, 40, 48. . . Multiples of 8: 8, 16, 24, 32, 40, 48. . . Multiples of 12: 12, 24, 36, 48. . . Multiples of 12: 12, 24, 36, 48. . . LCM is 24 LCM is 24 What is the LCM of 5 and 15? What is the LCM of 5 and 15? Multiples of 5: 5, 10, 15, 20, 25, 30, 35. . . Multiples of 5: 5, 10, 15, 20, 25, 30, 35. . . Multiples of 15: 15, 30 . . . Multiples of 15: 15, 30 . . . LCM is 15 LCM is 15 Prime Factorization – An expression of a whole number as a Prime Factorization – An expression of a whole number as a product of prime numbers. product of prime numbers. Ex. 2 x 2 x 2 x 5 = 40 Ex. 2 x 2 x 2 x 5 = 40 2 x 2 x 2 x 5 can also be written as 23 x 5 2 x 2 x 2 x 5 can also be written as 23 x 5 23 power Exponent 23 power base Exponent base Exponent – The number of times a base number is multiplied Exponent – The number of times a base number is multiplied by itself. by itself. 52 = 5 x 5 = 25 34 = 3 x 3 x 3 x 3 = 81 52 = 5 x 5 = 25 34 = 3 x 3 x 3 x 3 = 81 You can find the prime factorization of a number using a You can find the prime factorization of a number using a factor tree. factor tree. Factor Tree – A diagram used to decompose a composite Factor Tree – A diagram used to decompose a composite number into its prime factorization. number into its prime factorization. 40 4 x 10 2 x 2x 2 x 5 40 40 5 x 8 2 x 4 4 x 10 2 x 2x 2 x 5 40 5 x 8 2 x 4 2 x 2 Prime factorization of 40 is 23 x 5. 2 x 2 Prime factorization of 40 is 23 x 5. Finding the GCF using Factor Trees Finding the GCF using Factor Trees Step 1: Complete factor trees for each number. Step 1: Complete factor trees for each number. Step 2: Write down the prime factors they have in common. Step 2: Write down the prime factors they have in common. Step 3: Multiply the common prime factors to get the GCF. Step 3: Multiply the common prime factors to get the GCF. What is the GCF of 36 and 24? What is the GCF of 36 and 24? 36 4 24 x 9 4 2 x 2x 3 x 3 x 36 6 2 x 2x 2 x 3 x 9 4 2 x 2x 3 x 3 2x2x3x3 2x2x2x3 2x2x2x3 GCF = 12 What is the GCF of 55 and 66? 2 x 2 x 3 = 12 66 6 GCF = 12 x 55 11 5 66 x 11 6 2 x 3 x 11 2 x 3 5 x 11 2 x 3 x 11 6 What is the GCF of 55 and 66? 55 x 11 x 2 x 2x 2 x 3 2x2x3x3 2 x 2 x 3 = 12 5 4 24 5 x 11 GCF = 11 2 x 3 x 11 GCF = 11 Finding the LCM using Prime Numbers Finding the LCM using Prime Numbers Step 1: Write the numbers you are trying to find the LCM. Step 1: Write the numbers you are trying to find the LCM. Step 2: Draw a line under the numbers and down the left side. Step 2: Draw a line under the numbers and down the left side. Step 3: Beginning with the smallest prime number that will divide into a number, divide. Step 3: Beginning with the smallest prime number that will divide into a number, divide. Step 4: Continue dividing by the smallest prime number until both columns end in a 1. Step 4: Continue dividing by the smallest prime number until both columns end in a 1. Step 5: Multiply all the prime numbers on the left side. Step 5: Multiply all the prime numbers on the left side. What is the LCM of 36 and 24? 36 24 What is the LCM of 36 and 24? 36 24 2 18 12 2 18 12 2 9 6 2 9 6 2 9 3 2 9 3 3 3 1 3 3 1 3 1 3 1 2 x 2 x 2 x 3 x 3 = 72 2 x 2 x 2 x 3 x 3 = 72 LCM = 72 LCM = 72 What is the LCM of 22 and 66? 22 66 What is the LCM of 22 and 66? 22 66 2 11 33 2 11 33 3 11 11 3 11 11 11 1 1 11 1 1 2 x 3 x 11 = 66 LCM = 66 2 x 3 x 11 = 66 LCM = 66 Finding the LCM using Factor Trees Finding the LCM using Factor Trees Step 1: Complete factor trees for each number. Step 1: Complete factor trees for each number. Step 2: Write down the prime factors they have in common. Step 2: Write down the prime factors they have in common. Step 3: Write down any remaining prime numbers. Step 3: Write down any remaining prime numbers. Step 4: Multiply the numbers to get the LCM. Step 4: Multiply the numbers to get the LCM. What is the LCM of 36 and 24? 36 What is the LCM of 36 and 24? 36 4 x 9 24 4 2 x 2x 3 x 3 x 6 2 x 2x 2 x 3 x 9 4 2 x 2x 3 x 3 2x2x3x3 2x2x2x3 2x2x2x3 What is the LCM of 22 and 66? 22 x 11 LCM = 72 6 x 11 2 x 11 2 x 11 2 x 3 x 11 2 x 3 x 11 LCM = 66 x 2 x 3 2 x 11 2 x 11 x 3 = 66 LCM = 72 66 6 2 x 3 2 x 11 x 3 = 66 6 2 x 2 x 3 x 2 x 3= 72 What is the LCM of 22 and 66? 22 66 x 2 x 2x 2 x 3 2x2x3x3 2 x 2 x 3 x 2 x 3= 72 2 4 24 LCM = 66 11