DOCSLIB.ORG

Divisibility Rules Workbook

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Divisibility Rules Workbook Name _____________________ Divisibility A number is divisible by another number if it is a true multiple of the other number (or if the division problem does not have a remainder. 24 is divisible by 3 because 24 ÷ 3 = 8 with no remainder. 25 is not divisible by 3 because it would have a remainder of 1! This first section of this booklet will help you to learn the divisibility rules that you will need and use every day. The numbers are all less than 169 because using the Fraction Actions method, you will never end up with numbers greater than 169 that are not already finished numbers. The last section of the book will help you with a few rules that you will need to know for NC EOG testing purposes. Divisibility Rules Divisibility Rule for 2 When the last digit in a number is 0, 2, 4, 6, or 8, the number is divisible by 2 Divisibility Rule for 5 When the last digit in a number is 0 or 5, the number is divisible by 5 Divisibility Rule for 3 When the sum of the digits is a multiple of 3, the number is divisible by 3 Divisibility Rule for 7 Double the last digit in a number and subtract it from the rest of the number. If the result is a known multiple of 7, then the number is divisible by 7. If the result is not known, repeat the rule with the new number! Divisibility Rule for 11 When the sum of the odd digits is equal to (or 11 more than) the sum of the even digits, the number is divisible by 11 Divisibility Rule for 13 Nine times the last digit in a number and subtract it from the rest of the number. If the result is a known multiple of 13, then the number is divisible by 13. If the result is not known, repeat the rule with the new number! Divisibility Rule for 4 When the last 2 digits in a number are a multiple of 4, the number is divisible by 4 Divisibility Rule for 6 When a number is divisible by 2 and 3, the number is divisible by 6 Divisibility Rule for 9 When the sum of the digits is a multiple of 9, the number is divisible by 9 Divisibility Rule for 10 When the last digit in a number is 0, the number is divisible by 10 © 2009 – Norm Mitchell (Math6.org) – All Rights Reserved Freely reproducible for “non-profit” educational purposes – visit http://www.math6.org/legal.htm for more details concerning “non-profit”. Divisibility (2 and 5) Lesson Box Models The divisibility rule for 2 and 5 ask you to look at the 2, 5 80 last digit of the number. 80 ends with 0. According to the rules, numbers that If the last digit is 0, 2, 4, 6, or 8 the number is divisible end with 0 are divisible by 2 and 5. by 2. If the last digit is 0 or 5, the number is divisible by 5. 2 158 158 ends with 8. 8 is part of the 2 rule. Ø 151 151 ends with 1. 1 is not part of either rule, so 151 is not divisible by 2 or 5. 5 215 215 ends with 5. Twenty-five is divisible by 5! Apply the divisibility rules for 2 and 5. If the number is divisible by 2 or 5, write which ever answer is true. You do not need to write both answers. (example: 110 is divisible by both 2 and 5, you may answer 2) If the number is not divisible by these rules, enter Ø. ________ 160 ________ 145 ________ 107 ________ 37 ________ 41 ________ 80 ________ 136 ________ 115 ________ 20 ________ 156 ________ 158 ________ 133 ________ 120 ________ 137 ________ 59 ________ 151 ________ 124 ________ 35 ________ 105 ________ 34 ________ 76 ________ 78 ________ 141 ________ 24 ________ 112 ________ 25 ________ 106 ________ 159 ________ 23 ________ 162 ________ 108 ________ 36 ________ 44 ________ 68 ________ 61 ________ 66 ________ 103 ________ 111 ________ 167 ________ 100 ________ 87 ________ 73 ________ 157 ________ 67 ________ 96 ________ 72 ________ 130 ________ 46 ________ 54 ________ 30 ________ 29 ________ 154 ________ 74 ________ 89 ________ 99 [use Key 1 to check your answers] Divisibility (3) Lesson Box Models The divisibility rule for 3 asks you to find the sum of 3 138 the digits. 1 + 8 = 9 (or 1 + 3 + 8 = 12) If the sum of the digits is a multiple of 3, then the number is divisible by 3. Ø 58 257 5 + 8 = 13 2 + 5 + 7 = 14. 14 is not a multiple of 3. 257 is not divisible by 3. 3 126 342 1 + 2 = 3 (or 1 + 2 + 6 = 9) 3 + 4 + 2 = 9. 9 is a multiple of 3. 342 is divisible by 3. Ø 101 To cast out 3’s don’t add any 3, 6 or 9’s. 1 + 1 = 2 342 … 4 + 2 = 6 268 … 2 + 8 = 10 692 … 2 = 2 Apply the divisibility rule for 3. If the number is divisible by 3, write 3. If the number is not divisible by 3, enter Ø. ________ 47 ________ 164 ________ 54 ________ 156 ________ 108 ________ 70 ________ 68 ________ 137 ________ 55 ________ 117 ________ 122 ________ 77 ________ 92 ________ 107 ________ 106 ________ 140 ________ 61 ________ 83 ________ 87 ________ 31 ________ 169 ________ 148 ________ 141 ________ 78 ________ 41 ________ 51 ________ 145 ________ 56 ________ 96 ________ 74 ________ 111 ________ 46 ________ 90 ________ 155 ________ 19 ________ 60 ________ 64 ________ 44 ________ 72 ________ 124 ________ 86 ________ 81 ________ 75 ________ 167 ________ 36 ________ 100 ________ 131 ________ 159 ________ 134 ________ 95 ________ 38 ________ 119 ________ 166 ________ 69 ________ 59 ________ 14 ________ 26 ________ 50 ________ 40 ________ 29 [use Key 2 to check your answers] Divisibility Drills 2, 3 and 5 Apply the divisibility rules for 2, 3 and 5. If the number is divisible by 2, 3 or 5, write which ever answer is true. You do not need to write all of the answers. (example: 110 is divisible by both 2 and 5, you may answer 2) If the number is not divisible by these rules, enter Ø. ________ 103 ________ 75 ________ 65 ________ 30 ________ 139 ________ 24 ________ 112 ________ 28 ________ 105 ________ 123 ________ 53 ________ 143 ________ 64 ________ 26 ________ 136 ________ 49 ________ 90 ________ 164 ________ 140 ________ 96 ________ 50 ________ 152 ________ 111 ________ 104 ________ 125 ________ 31 ________ 81 ________ 145 ________ 71 ________ 113 ________ 169 ________ 107 ________ 95 ________ 131 ________ 129 ________ 19 ________ 165 ________ 149 ________ 29 ________ 124 ________ 36 ________ 127 ________ 137 ________ 84 ________ 109 ________ 141 ________ 32 ________ 63 ________ 100 ________ 14 [use Key 3 to check your answers] Apply the divisibility rules for 2, 3 and 5. If the number is divisible by 2, 3 or 5, write which ever answer is true. You do not need to write all of the answers. (example: 110 is divisible by both 2 and 5, you may answer 2) If the number is not divisible by these rules, enter Ø. ________ 55 ________ 48 ________ 98 ________ 40 ________ 83 ________ 119 ________ 155 ________ 130 ________ 151 ________ 157 ________ 51 ________ 106 ________ 116 ________ 135 ________ 27 ________ 17 ________ 70 ________ 25 ________ 117 ________ 42 ________ 120 ________ 156 ________ 160 ________ 62 ________ 34 ________ 101 ________ 41 ________ 133 ________ 121 ________ 72 ________ 153 ________ 146 ________ 159 ________ 148 ________ 45 ________ 44 ________ 134 ________ 167 ________ 122 ________ 79 ________ 162 ________ 161 ________ 67 ________ 168 ________ 126 ________ 154 ________ 12 ________ 110 ________ 85 ________ 23 [use Key 4 to check your answers] Divisibility (7) Lesson Box Models • Double the last digit in a number and subtract it from the rest of the number. Ø 138 • If the result is a known multiple of 7, then the number is divisible by 7. 8*2 = 16 ; 16-13 = 3 ; 138 is not divisible by 7 • If the result is not known, repeat the rule with the new number! 7 168 257 8*2 = 16 ; 16-16 = 0 ; 168 is divisible by 7 7 * 2 = 14. 25 – 14 = 11. 257 is not divisible by 7. 7 133 343 3*2 = 6 ; 13-6 = 7 ; 133 is divisible by 7 3 * 2 = 6. 34 – 6 = 28. 343 is divisible by 7. Ø 101 49 1*2 = 2 ; 10-2 = 8 ; 101 is not divisible by 7 9 * 2 = 18. 18 – 4 = 14. 49 is divisible by 7. (notice you can subtract the first digits from the doubled last if you need to!) Apply the divisibility rule for 7. If the number is divisible by 7, write 7. If the number is not divisible by 7, enter Ø. ________ 46 ________ 117 ________ 139 ________ 66 ________ 70 ________ 55 ________ 161 ________ 140 ________ 31 ________ 69 ________ 111 ________ 153 ________ 35 ________ 63 ________ 119 ________ 49 ________ 132 ________ 105 ________ 13 ________ 78 ________ 141 ________ 28 ________ 93 ________ 110 ________ 130 ________ 156 ________ 88 ________ 121 ________ 143 ________ 95 ________ 15 ________ 155 ________ 54 ________ 21 ________ 136 ________ 60 ________ 166 ________ 72 ________ 23 ________ 123 ________ 133 ________ 59 ________ 167 ________ 17 ________ 107 ________ 92 ________ 160 ________ 45 ________ 168 ________ 145 ________ 30 ________ 102 ________ 43 ________ 68 ________ 41 ________ 152 ________ 142 ________ 83 ________ 149 ________ 124 [use Key 5 to check your answers] Divisibility Drills 2, 3, 5, and 7 Apply the divisibility rules for 2, 3, 5 and 7.
Load more

Recommended publications
  • Divisibility Rules Pathway 1 OPEN-ENDED
    Divisibility Rules Pathway 1 OPEN-ENDED You will need I wonder if 72 is divisible by 9. • base ten blocks (optional) • a calculator Rememberemember • If a number is We know that 72 is divisible by 2, 3, 6, and 9 because you can divisible by another make groups of 2, 3, 6, or 9 and have nothing left over. number, when you divide them the For example, if you model 72 with blocks, you can make answer is a whole 8 groups of 9 and there are none left over. number. e.g., 72 4 6 5 12, so 72 is divisible by 6 (and by 12). 72 is divisible by 9 We know that 72 is not divisible by 5 or 10. This is because if you make groups of 5 or 10, there would always be some left over. A long time ago, people discovered shortcut rules for deciding whether numbers are divisible by 2, 3, 5, 6, 9, and 10. Each rule is one of these types: – If the ones digit of a number is ▲, the number is divisible by ❚. – If the sum of the digits of a number is divisible by ▲, the number is divisible by ❚. – If a number is divisible by both ▲ and ●, then it is also divisible by ❚. Note: Depending on the rule, the grey shape could represent 1 or more digits or a word. For example: – If the ones digit of a number is ▲, the number is divisible by ❚. If the ones digit of a number is even, the number is divisible by 2.
    [Show full text]
  • Divisibility Rules and Their Explanations
    Divisibility Rules and Their Explanations Increase Your Number Sense These divisibility rules apply to determining the divisibility of a positive integer (1, 2, 3, …) by another positive integer or 0 (although the divisibility rule for 0 says not to do it!)1. Explanations of the divisibility rules are included because understanding why a rule works helps you (1) to remember it and to use it correctly and (2) much more importantly, to deepen your knowledge of how numbers work. Some divisors have several divisibility rules; to limit the scope of this document, in most cases, only one divisibility rule is given for a divisor. A summary of the divisibility rules is included at the end. The explanations for these divisibility rules are divided into the following categories: x x x Last Digits 2 5 10 Modular Arithmetic 3 9 11 Composite Number Composites Prime Number Primes Other 0 1 00 11 22 33 44 55 66 77 88 99 1100 1111 1122 CCoommppoossiiitteess PPrriimmeess Before we get to the divisibility rules, let’s establish some groundwork required for the Last Digits and Modular Arithmetic explanations: Foundations: 1. Any integer can be rewritten as the sum of its ones + its tens + its hundreds + …and so on (for simplicity, we will refer to an integer’s ones, tens, hundreds, etc. as its “parts”). For example, the integer 3784 can be rewritten as the sum of its parts: 3000 + 700 + 80 + 4. 2. Dividing each of an integer’s parts by a divisor and summing the results gives the same result as dividing the integer as a whole by the divisor.
    [Show full text]
  • Week 4 of Virtual Guru Math Suhruth Vuppala Let’S Review What We Learned Last Time What Is the Area of Something?
    Welcome to Week 4 of Virtual Guru Math Suhruth Vuppala Let’s Review What We Learned Last Time What is the area of something? ● The area of a shape is how much space the shape takes up What is the perimeter of something? ● The perimeter of a shape is the length of the whole outside line or border of the shape ● The perimeter can usually be found by adding up all the side lengths of a shape How do you find the area of a square ● To find the area of the square you take one of the side lengths and multiply it by itself ● Let’s take the example of the square from two slides before. It has a side length of 5 centimeters, so you do 5x5 and you get the area ● What is the area of the square? How do you find the area of a rectangle? ● To find the area of a rectangle, you take the length of the top/bottom segments and multiply it by the length of the left/right segments ● Let’s use the example below. It has a right side length of 3 centimeters and a bottom side length of 8 centimeters. That means the area is 3x8 centimeters ● What is the area? How do you find the perimeter of a square? ● To find the perimeter of a square, you have to take one of the side lengths and multiply it by 4 ● Let’s again use the example from two slides before. It has a side length of 5 centimeters, so 5x4 is the perimeter of the square ● What is the perimeter? How do you find the perimeter of a rectangle? ● The perimeter of a rectangle can be found by doing 2 times the sum of the left/right and top/bottom ● In the example below, the right side length is 3 centimeters and the bottom side length is 8 centimeters.
    [Show full text]
  • Divisibility Rules © Owl Test Prep 2015
    Divisibility Rules © Owl Test Prep 2015 It’s my sincere hope that one of your elementary school teachers taught you these rules. If not, we can fix that. The rules can be broken down into 5 groups: 2, 4, and 8: Powers of 2 3 and 9: Multiples of 3 5 and 10: Multiples of 5 6: The mother of all divisibility rules However we can use the rules below to construct rules for lots of composite numbers. I’ll point out a couple of examples as we go along. Divisibility Rules for 2, 4, and 8 1) A number is divisible by 2 if a) The last digit is divisible by 2. For example 2,106 is divisible by two because the last digit, 6, is divisible by 2, but 2,119 is not divisible by 2 because 9 is not divisible by two. 2) A number is divisible by 4 if a) The last two digits form a number that is divisible by 4. For example 3,436 is divisible by 4 because 36 is divisible by 4, but 3,774 is not divisible by 4 because 74 is not divisibly by 4. Note that if the digit in the ten’s place is a zero the rule still applies; 104 and 2,308 are both divisible by 4. b) It is divisible by 2 twice. For example, , and 4, so 256 is divisible by 4, but 170 is not divisible by 4 because 170 , and 85 is not divisible by 2. Thus, 170 is only divisible by 2 once, so it is not divisible by 4.
    [Show full text]
  • Divisiblity Rules F (Taught).Notebook 1 August 07, 2013
    Divisiblity Rules F (taught).notebook August 07, 2013 Bell Ringer Name the decimal place indicated. Then round to that decimal place. Divisibility Rules 1. 67,149 2. 1.3446 3. 25.8 You will be able to . Divisiblity A number is divisible by another if the remainder • Determine whether a number is divisible by 2, is 0 when you divide. (In other words, if you get a 3, 5, 6, 9, or 10. whole number, not a decimal.) Ex: Divisible or not? a. 34 by 2 b. 10 by 4 c. 737 by 5 Divisibility Rule for 2 Example: Determine whether the following A number is divisible by 2 if it ends in 0, 2, 4, 6, numbers are divisible by 2. or 8. a. 542 b. 647 c. 9,001 d. 46 1 Divisiblity Rules F (taught).notebook August 07, 2013 Divisibility Rule for 5 Example: Determine whether the following A number is divisible by 5 if it ends in 0 or 5. numbers are divisible by 5. a. 75 b. 439 c. 3,050 d. 5 Divisibility Rule for 10 Example: Determine whether the following A number is divisible by 10 if it ends in 0. numbers are divisible by 10. a. 3005 b. 250 c. 622 d. 89,620 Example: State whether each number is divisible Your Turn: State whether each number is by 2, 5, or 10. (More than one answer is divisible by 2, 5, or 10. (More than one answer is possible!) possible!) a. 288 b. 300 c. 4,805 1. 50 2. 1,015 3.
    [Show full text]
  • Stupid Divisibility Tricks 101 Ways to Stupefy Your Friends Appeared in Math Horizons November, 2006
    Stupid Divisibility Tricks 101 Ways to Stupefy Your Friends Appeared in Math Horizons November, 2006 Marc Renault Shippensburg University Mathematics Department 1871 Old Main Road Shippensburg, PA 17013 [email protected] http://webspace.ship.edu/msrenault/divisibility/ 1 Introduction When is the last time this happened to you? You are stranded on a deserted island without a calculator and for some reason you must determine if 67 is a divisor of 95733553; furthermore, a coconut recently fell on your head and you have completely forgotten how to perform long division. Of course the above scenario would never happen (we all carry around calculators) but it's good to know that if we should ¯nd ourselves in a similar situation there is an easy divisibility rule for 67: remove the two rightmost digits from the number (in our case, 53), double them (106) and subtract that from the remaining digits (957335¡106 = 957229); the original number is divisible by 67 if and only if the resulting number is divisible by 67. If the resulting number is not obviously divisible by 67 we can repeat the process until we get a number that clearly is or is not a multiple of 67. In the above example, we get the following. 1 9573355==3 { 106 95722==9 { 58 951==4 { 28 67 Thus we conclude that in fact 95733553 is a multiple of 67. This article has two aims. First, to identify six categories of tests that most divisibility tricks fall into, and second, to provide an easy divisibility test for each number from 2 { 102 (thus the \101 Ways..." in the title).
    [Show full text]
  • Number Theory
    2015 – 2016 Log1 Contest Round 2 Theta Number Theory Name: __________________ 4 points each 1 Find the number of positive integral divisors of the number 4. 2 Find the sum of the integers from 1 to 63, inclusive. 3 How many of the following five integers are divisible by 7?: 6, 42, 245, 3122, 43659 4 What fraction of the 50 least positive perfect squares are also perfect cubes? 5 What fraction of the 50 least positive perfect cubes are also perfect squares? 5 points each 6 Find the sum of the positive integral divisors of the number 4. 7 What is the greatest common factor of 3458 and 3850? 8 What is the least common multiple of 3458 and 3850? 9 Find the two-digit number at the end of 12373 when it is expanded. 10 How many positive integral factors of 2016 are divisible by 3? 6 points each 11 When expanded, in how many consecutive zeros does 23! 24! end? 12 In Java programming, there exists a command on integers called Integer Remainder Division, symbolized by %. For example, 20%1 0 since 20 leaves a remainder of 0 when divided by 1; also, 8%3 2 since 8 leaves a remainder of 2 when divided by 3. What would be the output on the Java command 3871183%1967 ? 13 How many ordered pairs xy, of positive integers satisfy the equation 19xy 29 12673? 14 The sum of two positive prime numbers is 99. Find the product of these two prime numbers. 15 “Take a k-digit integer n, where k 2 .
    [Show full text]
  • PMATH 340 Lecture Notes on Elementary Number Theory
    PMATH 340 Lecture Notes on Elementary Number Theory Anton Mosunov Department of Pure Mathematics University of Waterloo Winter, 2017 Contents 1 Introduction . .3 2 Divisibility. Factorization of Integers. The Fundamental Theorem of Arithmetic . .5 3 Greatest Common Divisor. Least Common Multiple. Bezout’s´ Lemma. .9 4 Diophantine Equations. The Linear Diophantine Equation ax + by = c ........... 15 5 Euclidean Algorithm. Extended Euclidean Algorithm . 18 6 Congruences. The Double-and-Add Algorithm . 24 7 The Ring of Residue Classes Zn .................. 29 8 Linear Congruences . 31 ? 9 The Group of Units Zn ....................... 33 10 Euler’s Theorem and Fermat’s Little Theorem . 36 11 The Chinese Remainder Theorem . 38 12 Polynomial Congruences . 41 13 The Discrete Logarithm Problem. ? The Order of Elements in Zn .................... 45 14 The Primitive Root Theorem . 50 15 Big-O Notation . 53 16 Primality Testing . 56 16.1 Trial Division . 57 16.2 Fermat’s Primality Test . 58 16.3 Miller-Rabin Primality Test . 61 17 Public Key Cryptosystems. The RSA Cryptosystem . 62 18 The Diffie-Hellman Key Exchange Protocol . 67 19 Integer Factorization . 69 1 19.1 Fermat’s Factorization Method . 70 19.2 Dixon’s Factorization Method . 72 20 Quadratic Residues . 75 21 The Law of Quadratic Reciprocity . 81 22 Multiplicative Functions . 86 23 The Mobius¨ Inversion . 91 24 The Prime Number Theorem . 95 25 The Density of Squarefree Numbers . 96 26 Perfect Numbers . 101 27 Pythagorean Triples . 104 28 Fermat’s Infinite Descent. Fermat’s Last Theorem . 105 29 Gaussian Integers . 110 30 Fermat’s Theorem on Sums of Two Squares . 120 31 Continued Fractions . 124 32 The Pell’s Equation .
    [Show full text]
  • Divisibility Rules
    Divisibility Rules A number is divisible by another number if it divides evenly without a remainder. For example, 8 is divisible by 4 since 8 + 4 = 2 RO . The following divisibility rules apply to whole numbers. Divisor: 2 Divisibility Rule: Any whole number that ends in 0, 2, 4, 6, or 8 is divisible by 2. (i.e. Any even number is divisible by 2). Example: 78558 is divisible by 2 since the last digit is 8. Non-Example: 78559 is not divisible by since the last digit is 9. Divisor: 3 Divisibility Rule: If the sum of the digits is divisible by 3, then the whole number is also divisible by 3. Example: 1539 is divisible by 3 since 1 + 5 + 3 + 9 18 and 18 is divisible by 3. = Non-Example: 1540 is not divisible by 3 since 1 + 5 + 4 + 0 = 10 and 10 is not divisible by 3. ©Tutoring and Learning Centre, George Brown College, 2020 | www.georgebrown.ca/tlc Divisor: 4 Divisibility Rule: If the last two digits of the whole number form a two-digit number that is divisible by 4, then the whole number is also divisible by 4. Example: 19236 is divisible by 4 since the last two digits, 36, form a number that is divisible by 4. Non-Example: 19237 is not divisible by 4 since the last two digits, 37, form a number that is not divisible by 4. Divisor: 5 Divisibility Rule: Any whole number that ends in 0 or 5 is divisible by 5. Example: 87235 is divisible by 5 since the number ends in 5.
    [Show full text]
  • 6Thth Grade Math
    MATH TODAY Grade 6, Module 2, Topic D 2014/2015 th 6th Grade Math Focus Area Topic D: Module 2: Arithmetic Operations Including Divisibility Rules Division of Fractions Students use these rules to make determining if a larger Math Parent Letter number is divisible by 2, 3, 4, 5, 8, 9, or 10 easier. Divisibility Rule for 2: If and only if its last digit is even. This document is created to give parents and students a better understanding of the math concepts found in Engage New Divisibility Rule for 3: If the sum of the digits is divisible by York, which correlates with the California Common Core 3, then the number is divisible by 3. Standards. Module 2 of Engage New York introduces the concepts of Arithmetic Operations Including Division of Divisibility Rule for 4: If and only if its last two digits are a Fractions. number divisible by 4. Divisibility Rule for 5: If and only if its last digit is 0 or 5. Divisibility Rule for 8: If and only if the last three digits are a number divisible by 8. Divisibility Rule for 9: If the sum of the digits is divisible by Focus Area Topic D: 9, then the number is divisible by 9. Divisibility Rule for 10: If and only if its last digit is 0. Words to Know: Factor: 2 x 3 = 6 Focus Area Topic D: Factor Factor Greatest Common Factor Example Problems and Answers Multiples: Multiples of 6: 6, 12, 18, 24, 30… GCF 30 and 50 Multiples of 7: 7, 14, 21, 28, 32… Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 Factors of 50: 1, 2, 5, 10, 25, Greatest Common Factor (GCF): The greatest of 50 the common factors of two or more numbers.
    [Show full text]
  • Divisibility Rules!
    Divisibility Rules! Investigating Divisibility Rules Learning Goals Key Term In this lesson, you will: divisibility rules Formulate divisibility rules based on patterns seen in factors. Use factors to help you develop divisibility rules. Understanding relationships between numbers can save you time when making calculations. Previously, you worked with factors and multiples of various numbers, and you determined which numbers are prime and composite by using the Sieve of Eratosthenes. By doing so, you determined what natural numbers are divisible by other natural numbers. In this lesson, you will consider patterns for numbers that are divisible by 2, 3, 4, 5, 6, 9, and 10. What type of patterns do you think exist between these numbers? Why do you think 1 is not a part of this list? © 2011 Carnegie Learning 1.4 Investigating Divisibility Rules • 35 © 2011 Carnegie Learning 1.4 Investigating Divisibility Rules • 35 Problem 1 Students explore the divisibility of numbers by 2, 5 and 10. Problem 1 Exploring Two, Five, and Ten They will list multiples of given numbers and notice 1. List 10 multiples for each number. all multiples of 2 are even numbers, all multiples of 5 have Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 a last digit that ends in a 0 or Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 5, and all multiples of 10 are also multiples of both 2 and 5. Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 Students will write divisibility 2.
    [Show full text]
  • Divisibility Rules Justified Grades 6-8
    Divisibility Rules Justified Grades 6-8 Standards: 7 MR3.3 Develop generalizations of the results obtained and the strategies used and apply them to new problem situations. 7.EE.1 Use properties of operations to generate equivalent expressions. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Materials: Class set of properties cards (pages 7-11) for students to use in pairs (or small groups) Partial notes handout (pages 12-13), one per student Introduction: “Divisibility rules can be quick shortcuts for factoring multi-digit numbers. Today we are going to examine why they work using proofs with variables. In middle school, you are not expected to be able to write proofs like this, but you will be able to justify each step, working with your partner (or small group).” Ex 1 (I Do): Divisibility rule for 3 A number is divisible by 3 iff (if and only if) the sum of its digits is a multiple of 3. The thousands digit times 1000, the Proof: hundreds digit times 100, the tens digit N N abcd Let be a four digit number such that = . times 10 plus the ones digit. N = a • 1000 + b •100 + c •10 + d Expanded notation N = a(999 +1) + b(99 +1) + c (9 +1) + d Decomposition N = 999 a + a + 99b + b + 9c + c + d Distributive Property of Multiplication Over Addition N = 999 a + 99 b + 9c + a + b + c + d Commutative Property of Addition N = 3• 333a + 3• 33b + 3• 3c + (a + b + c + d ) Decomposition In this step, we decompose into factors to show parts that are divisible by 3.
    [Show full text]