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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. 156 Divisibility Rules, Pathway 1 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd. • Create divisibility rules for the numbers below. For each, use one of the 3 types of rules listed on the previous page. You might use the same type of rule more than once. divisibility rule for 2 divisibility rule for 6 If the last digit of a number is even, the number If a number is divisible by both 2 and 3, then it is is divisible by 2. also divisible by 6. divisibility rule for 3 divisibility rule for 9 If a number is divisible by both 2 and 3, then it is If the sum of the digits of a number is divisible also divisible by 6. by 9, the number is divisible by 9. divisibility rule for 5 divisibility rule for 10 If the last digit of a number is 0 or 5, the If the last digit of a number is 0, the number is number is divisible by 5. divisible by 10. • Choose 2 of the rules you wrote above and explain why each makes sense. e.g., The test for 10: e.g., The test for 6: If a number can be grouped in 10s, when you A number that can be grouped in 6s can also be model it with base ten blocks there would be no arranged in pairs (each group of 6 is 3 pairs) ones, so 0 would be in the ones place. with 0 left over, which makes it a multiple of, or divisible by 2. It can also be grouped in 3s (each group of 6 is 2 groups of 3), which makes it a multiple of, or divisible by 3. 157 Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Divisibility Rules, Pathway 1 Divisibility Rules Pathway 1 GUIDED At a school bake sale, cookies were in packages of 3. Keifer said that there were 428 cookies in total. Jane said there couldn’t be 428 cookies in total. Jane added the digits 4 1 2 1 8 5 14. Then she said that, Remembeemember since 14 is not a multiple of 3, there could not be 428 cookies. She used the divisibility rule for 3 to figure out that 428 was not • If a number is divisible by another divisible by 3. number, the result will be a whole Divisibility by 3 Rule number when you If the sum of the digits of a number is divisible by 3, then the number divide. is divisible by 3. e.g., 72 4 6 5 12, so 72 is divisible by You can look at the pattern in the sum of the digits for multiples 6 (and by 12). of 3 from 0 to 57 to see why this rule works: Multiples of 3 Multiple of 3 036912 15 1821 24 27 Sum of digits 0369369369 Multiple of 3 30 33 36 39 42 45 48 51 54 57 Sum of digits 369126 9126 912 Notice the following: • As you go from one multiple of 3 to the next, the sum of the digits increases or decreases by a multiple of 3. So, each sum continues to be divisible by 3. • Once you have added 3 three times (or added 9) in the row of multiples, the ones digit goes down 1 and the tens digit goes up 1 (since 9 5 10 2 1). That means the sum of the digits is the same as the earlier sum. For example, from 15 to 24, 9 is added. The sum of the digits for 24 is 6. This is the same sum as for 15. 158 Divisibility Rules, Pathway 1 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd. Here are some other divisibility rules that can be used to figure out if 428 cookies can be in packages of 2, 5, 6, 9, or 10 with none left over. Divisibility by 9 Rule Divisibility by 2 Rule If the sum of the digits of a number is If the ones digit of a number is even, divisible by 9, the number is divisible by 9. the number is divisible by 2. Can 428 cookies be in packages of 9? Can 428 cookies be in packages of 2? 4 1 2 1 8 5 14, and 14 is not a multiple of 9, The digit 8 is even, so 428 cookies can so 428 cookies cannot be packaged in 9s. be packaged in 2s. Divisibility by 5 and 10 Rule Divisibility by 6 Rule If the ones digit of a number is 5 or 0, If the ones digit of a number is even and the number is divisible by 5. the sum of its digits is divisible by 3, then If the ones digit of a number is 0, the number is divisible by 6. the number is divisible by 10. Can 428 cookies be in packages of 6? Can 428 cookies be in packages of 5 or 10? 8 is even, but 4 1 2 1 8 5 14, which is The ones digit is 8, not 5 or 0, so 428 not divisible by 3, so 428 cookies cannot cookies cannot be packaged in 5s or 10s. be packaged in 6s. Try These 1. Use the divisibility rules to answer these questions about the numbers on the right. a) Which are divisible by 2? 258, 612, 490, 610 615 387 b) Which are divisible by 3? 258, 429, 387, 612, 369, 615 258 c) Which are divisible by 5? 490, 615, 610 610 490 d) Which are divisible by 6? 258, 612 369 e) Which are divisible by 9? 387, 612, 369 612 429 f ) Which are divisible by 10? 490, 610 2. Every answer for Question 1e) was also an answer for 1b), but not the other way around. Explain why. e.g., If a number can be grouped in 9s, each group of 9 can be grouped into 3 groups of 3; but you can't make 1 or 2 groups of 3 into a group of 9. 159 Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Divisibility Rules, Pathway 1 3. Write digits in the blanks to make numbers that will make the statements true. Complete each statement in 2 ways. a) 5 4 9 is divisible by 9. 5 9 4 is divisible by 9. b) 2 2 26 is divisible by 6. 1 0 26 is divisible by 6. c) 3 2 4 0 is divisible by 2. 3 2 4 0 is divisible by 5. d) 69 1 2 0 is divisible by 3 and 10. 69 4 2 0 is divisible by 3 and 10. 4. a) Complete this chart for multiples of 9. Multiple of 9 90 99 108 117 126 135 144 153 162 171180 189 Sum of digits 9 18 9 9 9 9 9 9 9 9 9 18 b) Describe the patterns in the multiples of 9. e.g., As you go up the multiples, the ones digit is 1 less and the tens digit is 1 more. Once the ones digit gets to 0, then the next ones digit is 9 and the tens digit doesn't change. c) What do you notice about the sums of the digits? e.g., The sum of the digits is usually 9 but sometimes 18. d) How do the patterns relate to the divisibility rule for 9: If the sum of the digits of a number is divisible by 9, the number is divisible by 9. e.g., The sum of the digits is 9 or 18, and both are divisible by 9. 5. Choose a divisibility rule for 2, 5, or 10. Explain why it makes sense. e.g., The test for 5: If you start with 5 and keep adding 5s, you hit every number that ends in 0 or 5, so the rule makes sense. 160 Divisibility Rules, Pathway 1 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.
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