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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. What do you notice? rules for 2, 5, and 10. All multiples of 2 are even and end in 0, 2, 4, 6, or 8. All multiples of 5 end in a 0 or a 5. All multiples of 10 end in a 0. Also, all multiples of 10 are also multiples of 2 and 5. Grouping Have students complete Divisibility rules are tests for determining whether one whole number is divisible by Questions 1 through 3 with another. A divisibility rule must work for every number. a partner. Then share the 3. Write a divisibility rule for 2, 5, and 10. Then, show an example that follows your rule. responses as a class. A natural number is …if Example divisible by Share Phase,

Questions 1 through 3 2 the ones digit is 0, 2, 4, 6, 186 is divisible by 2 because the • All numbers are divisible by or 8; the number is even. ones digit is 6. what number? • Are all numbers divisible by some other number?

Can you think of a number • 5 the ones digit is either 145 is divisible by 5 because the that is not divisible by any 0 or 5. ones digit is 5. other number? • What number is the © 2011 Carnegie Learning multiplicative identity?

Are there any shortcuts 10 the ones digit is 0. 630 is divisible by 10 because the • ones digit is a 0. you know for checking for divisibility? • If you listed 20 multiples for each number, would the 36 • Chapter 1 Factors, Multiples, Primes, and Composites same pattern emerge? • If a number is divisible by 10 is it also divisible by 5? • Can a divisibility rule be used • If a number is divisible by both 2 and 5, why must it be divisible by 10? on all numbers? © 2011 Carnegie Learning • If a number is divisible by 5 is it also divisible by 10?

36 • Chapter 1 Factors, Multiples, Primes, and Composites © 2011 Carnegie Learning a numberby3? write aruleforthedivisibilityof inorderdetermine apattern to you performthatmayhelp number, whatoperationcan Consider allofthedigits Questions 1and2 Share Phase, responses asaclass. a partner. Thenshare the Questions 1through 5with Have studentscomplete Grouping Calculator Materials rule usingacalculator. They thentesteachdivisibility divisibility rulesfor3,and6. andwrite look forpatterns given numbers,studentswill 2, 5,and10.Afteranalyzing given numbersare divisible by 3 anddeterminewhichofthe numbers thatare divisible by will beginbyusingalistof of numbersby3,and6.They Students explore thedivisibility Problem 2

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the number.) indicate whenanumberisdivisibleby3.(Hint:Considerthesumofdigits Analyze eachnumberthatisdivisibleby3.Then,writearuleinthetableshownto or 10. Place acheckintheappropriate columnforeachnumberthatisdivisibleby2,5,

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37 Share Phase, Questions 3 through 5 • If a number is divisible by 3. Circle numbers you think are divisible by 6 in the table you completed in Question 1. Explain your reasoning. both 2 and 3, why must it be The numbers I think are divisible by 6 are 300, 882, 1230, 3762, 42, 2784, and 3582. divisible by 6? These numbers are divisible by both 2 and 3, which are both factors of 6. So, I • If a number is divisible by 3, think that these numbers are also divisible by 6. is it also divisible by 6? • If a number is divisible by 6, is it also divisible by 3? 4. Analyze each number you circled that you think is divisible by 6. Write a rule to indicate when a number is divisible by 6 in the table shown. • If a number is divisible by 6,

why must it be divisible by 3? A number is …if Example divisible by

6 the number is divisible by 264 is divisible by 6 because it is both 2 and 3. divisible by 2 and divisible by 3.

5. Test the divisibility rules you wrote to indicate if a number is divisible by 3 or 6 by writing several three- or four-digit numbers that you think are divisible by 3 or 6. Then, use your calculator to determine if the numbers you wrote are divisible by 3 or 6. Answers will vary. © 2011 Carnegie Learning

38 • Chapter 1 Factors, Multiples, Primes, and Composites © 2011 Carnegie Learning responses asaclass. a partner. Thenshare the Questions 1through 3with Have studentscomplete Grouping rule usingacalculator. They thentestthedivisibility write adivisibilityrulefor9. and students lookforpatterns After analyzinggivennumbers, divisible by2,3,5,6,and10. which ofthegivennumbersare are divisibleby9anddetermine by usingalistofnumbersthat of numbersby9.Theybegin Students explore thedivisibility Problem 3

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or 10.ThecolumnforDivisibleby9iscompletedyou. Place acheckintheappropriate columnforeachnumberthatisdivisibleby2,3,5,6, rule for3.) divisible by9.(Hint:Usethesameclueyouwere givenwhenexploringthedivisibility Analyze thenumbersshowninlist.Write aruletoindicatewhennumberis Number A numberis

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39 Share Phase, Questions 1 through 3 • If a number is divisible by 3, 3. Test the divisibility rule you wrote to indicate if a number is divisible by 9 by writing several four- or five-digit numbers that you think are divisible by 9. Then, use your is it also divisible by 9? calculator to determine if the numbers you wrote are divisible by 9. • If a number is divisible by 9, Answers will vary. is it also divisible by 3? • If a number is divisible by 6, is it also divisible by 9? • If a number is divisible by 9, is it also divisible by 6? Problem 4 Exploring Four • If a number is divisible by 9, why must it be divisible by 3? Each number listed in the table is divisible by 4.

Numbers Divisible by 4 Problem 4 116 35,660 Students explore the divisibility of numbers by 4. They begin 1436 18,356 using a list of numbers that are 228 300,412 divisible by 4. After analyzing 2524 59,140 each number, students will look for patterns and write a 41,032 79,424 divisibility rule for 4. This rule involves the sum of the last 1. What pattern do you notice about each number? (Hint: Look at the number formed two digits of each number, so by the last two digits in each number.) they are given a hint. They then Each number is an even number, or each number is divisible by 2. test the divisibility rule using a The last two digits of each number also form a number that is divisible by 4. calculator.

2. Write a rule to tell when a number is divisible by 4. Grouping A number is …if Example Have students complete divisible by

Questions 1 through 3 with © 2011 Carnegie Learning a partner. Then share the the number formed by the last 316 is divisible by 4 4 responses as a class. two digits is divisible by 4. because 16 is divisible by 4.

40 • Chapter 1 Factors, Multiples, Primes, and Composites © 2011 Carnegie Learning • • Questions 1and 2 Share Phase, responses asaclass. a partner. Thenshare the Questions 1through 5with Have studentscomplete Grouping mystery number. rules, studentsidentifya clues andusingthedivisibility numbers. Givenaseriesof that are divisiblebygiven reasoning. Theycreate numbers numbers andexplaintheir test thedivisibilityofseveral rules theyhavecreated to Students usethedivisibility Problem 5 • • • • • Questions 1through 3 Share Phase, the lastdigit? what numberscouldnotbe If anumberisdivisibleby3, last digit? what numberscouldbethe If anumberisdivisibleby3, why mustitbedivisibleby2? If anumberisdivisibleby4, is italsodivisibleby4? If anumberisdivisibleby8, is italsodivisibleby8? If anumberisdivisibleby4, is italsodivisibleby2? If anumberisdivisibleby4, is italsodivisibleby4? If anumberisdivisibleby2,

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Answers determine ifthenumbersyouwrote are divisibleby4. several five-digit numbers that you think are divisible by 4. Then, use your calculator to Test the divisibility rule you wrote to indicate if a number is divisible by 4 by writing c. b. a. Explain yourreasoning. Determine ifeachnumberisdivisibleby3usingyourdivisibilityrule. c. b. a. Explain yourreasoning. Determine ifeachnumberisdivisibleby9usingyourdivisibilityrule.

No. 83,594 divisible Yes. 2109 divisible Yes. 597 27 Yes. 43,695 divisible Yes. 5814 divisible No. 748 and

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41 Share Phase, Questions 3 through 5 • If a number is a two digit 3. Fill in the missing digit for each number to make the sentence true. number, can you use any a. The number 10,5__2 is divisible by 6. divisibility rules to eliminate a The possible values are 1, 4, or 7. factor? b. The number 505__ is divisible by 4. • What multiples of 5 do not The possible values are 2 or 6. have a last digit that is a 5? • What are some numbers that c. The number 133,0__5 is divisible by 9. are both a multiple of 5 and The value is 6.

divisible by 3? 4. Rasheed is thinking of a mystery number. Use the following clues to determine his number. Explain how you used each clue to determine Rasheed’s number. Clue 1: My number is a two-digit number. Clue 2: My number is a multiple of 5, but does not end in a 5. Clue 3: My number is less than 60. Clue 4: My number is divisible by 3. Rasheed’s number is 30. Clue 1: The number is between 10 and 99 inclusive. Clue 2: Since the number is a multiple of 5, but not ending in 5, the number must end in a zero. The number could be 10, 20, 30, 40, 50, 60, 70, 80, or 90. Clue 3: Since the number is less than 60, it could be 10, 20, 30, 40, or 50. Clue 4: Since the number is divisible by 3, the number can only be 30.

5. Think of your own mystery number, and create clues using what you know about factors, multiples, and the divisibility rules. Give your clues to your partner. See if your partner can determine your mystery number! Answers will vary based on correct use of divisibility rules, factors, and multiples.

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