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. 6. What would you say to each person to explain why the divisibility rule will not work? Use examples to help you explain. “I think a number is divisible by 2 if the sum of its digits is even.”
e.g., The sum of the digits of 410 is 5, which is odd, and the sum of
the digits of 420 is 6, which is even, but they are both divisible by 2.
“I think a number is divisible by 6 if the sum of its digits is divisible by 6.”
e.g., The sum of the digits of 51 is 6 and 51 is not divisible by 6.
7. Compare Tyler’s divisibility by 6 rule at the right with this rule: If the ones digit of a number is even and the sum of its digits If a number is is divisible by 3, then the number is divisible by 6. divisible by 6, it is Do you agree with what Tyler is saying? Explain your thinking. divisible by both 2 and 3. Yes, e.g., Saying the ones digit is even is the same as saying it's
divisible by 2 and saying the sum of the digits is divisible by 3 is the
same as saying it's divisible by 3.
8. Think about Tyler’s divisibility by 6 rule. How could you use a similar idea to create a divisibility by 15 rule?
e.g., Combine the tests for 3 and 5.
9. Why do you think divisibility rules were more useful in the past than they are now? FYI e.g., Almost everyone has a calculator now. Knowing about divisibility rules can make it easier to determine if a number is prime or composite.
161 Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Divisibility Rules, Pathway 1 Prime Numbers and Perfect Squares Pathway 2 OPEN-ENDED
Part A You will need
Some numbers of tiles can be arranged in different rectangles. • square tiles Some numbers of tiles can be arranged in only 1 rectangle. • a calculator
I made 3 different rectangles with 12 tiles but only 1 rectangle with 11 tiles.
composite a whole number with more than 2 factors e.g., 6 is composite because 6 5 2 3 3 If tiles can be arranged in 2 or more different rectangles, and 6 5 1 3 6. that number of tiles is a composite number. prime For example, 12 is a composite number. a whole number with exactly 2 factors— If only 1 rectangle is possible, the number is a prime number. itself and 1 11 For example, is a prime number. e.g., 7 is prime The number 1 is neither prime nor composite. because its only factors are 7 and • Figure out which numbers from 2 to 50 are prime and 1 (7 5 7 3 1). which are composite. List them below.
prime: composite: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50
• Is the sum of 2 prime numbers usually a prime number? Explain.
No; e.g., Since you're usually adding two odds, the sum would be even and composite, but if one of the primes was 2, the sum could be prime.
• Can the product of 2 prime numbers be a prime number? Explain.
No, e.g., If a number is the product of primes, each prime is a factor in addition to the number itself and 1, so it cannot be prime
162 Prime Numbers and Perfect Squares, Pathway 2 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd. Part B If a number of tiles can be arranged in a square, that number of tiles is called a perfect square. perfect square a number that is the The side length of the square is the square root of the number product of a whole of tiles. number multiplied For example, 4 is a perfect square, and 2 is the square root of 4. by itself e.g., 9 is a perfect square because 3 3 3 5 9. 2 square root a number that you 2 2 4 can multiply by itself to get a given number • Complete the charts, using numbers from 1 to 200 that are perfect e.g., 4 is the square squares. Write the square root for each number. root of 16 because Squares and Square Roots 4 3 4 5 16.
Perfect squares 1 4 9 16 25 36 49
Square roots 1 2 3 4 5 6 7
Perfect squares 64 81 100 121 144 169 196
Square roots 8 9 10 11 12 13 14
• Can the sum of 2 perfect squares be a perfect square? If so, do you think this is always true? Explain, using examples.
Yes, e.g., 36 + 64 = 100 and 100 is a perfect square (10 x 10). No, e.g., 36 + 9 = 45 and 45 is not a perfect square. 16 + 16 = 32 and 32 is not a perfect square.
• Can the product of 2 perfect squares also be a perfect square? Explain, using examples.
Yes, e.g., 4 x 9 = 36 and 36 is a perfect square (6 x 6), 9 x 25 = 225 and 225 is a perfect square (25 x 25), and 4 x 25 = 100 and 100 is a perfect square (10 x 10).
163 Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Prime Numbers and Perfect Squares, Pathway 2 Prime Numbers and Perfect Squares Pathway 2 GUIDED
Some numbers are given special names because of the way You will need they can be written as products. • square tiles Prime and Composite Numbers • a calculator • 4 square tiles can be arranged in 2 different rectangles — a 2-by-2 rectangle and a 1-by-4 rectangle. That means you can write 4 as 2 3 2 or as 1 3 4, and the factors of 4 are 1, 2, and 4.
2 2 1 4 Since 4 can be arranged in more than 1 rectangle and it has more than 2 factors, 4 is a composite number. composite a whole number with • 3 square tiles can be arranged in only 1 rectangle — a 1-by-3 more than 2 factors 1 3 rectangle. That means you can write 3 as 3, and the factors e.g., 6 is composite of 3 are 1 and 3. because 6 5 2 3 3 and 6 5 1 3 6. prime a whole number with 1 3 exactly 2 factors — Since 3 can be arranged in only 1 rectangle and it has exactly itself and 1 2 factors, 3 is a prime number. e.g., 7 is prime because its only 1 • The number is neither a prime nor a composite because it has factors are 7 and only 1 factor. 1 (7 5 7 3 1). Perfect Squares and Square Roots perfect square a number that is the • 4 tiles can be arranged in a square with a side length of 2. product of a whole 9 tiles can be arranged in a square with a side length of 3. number multiplied by itself e.g., 9 is a perfect square because 3 3 3 3 5 9. 2 square root a number that you can multiply by 2 2 4 3 3 9 itself to get a given number That means the numbers 4 and 9 are perfect squares and e.g., 4 is the square their side lengths are called square roots. root of 16 because 2 is the square root of the perfect square 4. 4 3 4 5 16. 3 is the square root of the perfect square 9.
164 Prime Numbers and Perfect Squares, Pathway 2 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd. Try These 1. Circle the prime numbers.
49 101 201 89
2. Why is 5 the only prime number with a 5 in the ones place?
e.g., Every number other than 5 that ends in 5 can be grouped in more
than 1 group of 5; e.g., 15 = 3 x 5, 25 = 5 x 5, and 35 = 7 x 5.
3. Can a prime number be even? Explain your thinking.
Yes but only 2; e.g., All other even numbers are multiples of 2 so they
have more than 2 factors (1, 2, and the number itself are always 3 of
those factors).
4. Choose 2 composite numbers. Write each as a product of prime numbers. For example, 24 5 2 3 2 3 2 3 3.
e.g., 40 = 2 x 2 x 2 x 5
65 = 5 x 13
5. Suppose you want to figure out if 143 is a prime number and you already know that 2, 3, 5, and 7 are not factors of 143. a) Would you divide 143 by 9 to see if 9 is a factor? Explain.
No; e.g., If 3 is not a factor, 9 can't be.
b) Would you divide 143 by any numbers greater than 100 to see if they are factors? Explain your thinking.
No; e.g., 100 x 2 is already 200 which is too big.
c) Would you divide 143 by 11 to see if it is a factor? Explain.
Yes; e.g., 11 is a factor because 11 x 13 = 143, so 143 is not prime.
165 Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Prime Numbers and Perfect Squares, Pathway 2 6. Can the sum of 2 prime numbers be prime? Explain.
Yes; e.g., only if one of them is 2. If you add two primes, which are odd
except for the number 2, the sum is even, which means it's not a prime.
7. Circle each perfect square and write its square root.
49 81 90 121 141 169
7, 9, 11, 13 There are no perfect squares 8. Do you agree with what Karma says at the right? between 225 Explain your thinking. and 256.
Yes, e.g., 225 = 15 x 15 and 256 = 16 x 16 and there are no whole
numbers between 15 and 16.
9. Since 9 + 16 = 25, we know that the sum of 2 perfect squares can be a perfect square. a) Give another example of when the sum of 2 perfect squares is a perfect square.
e.g., 36 + 64 = 100 b) Is the sum of 2 perfect squares always a perfect square? Explain your thinking using an example.
No, e.g., 9 + 25 = 34, which is not a perfect square. c) Can the product of 2 perfect squares be a perfect square? Explain your thinking using an example.
Yes, e.g., 9 x 16 = 144 and 144 is 12 x 12.
10. a) Why do you think there are a lot fewer prime numbers FYI than composite numbers? Knowing about prime numbers will help you e.g., Every second number is even and a lot of the odd numbers know when a number has been factored as are multiples of 3 or 5, so a lot of numbers are composite. much as it can be. b) Why do you think there are a lot fewer perfect squares than other whole numbers?
e.g., Square numbers are 1, 4, 9, 16, and so on and they get farther
apart with many other whole numbers between them.
166 Prime Numbers and Perfect Squares, Pathway 2 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd. Factors and Multiples Pathway 3 OPEN-ENDED
You will need I arranged 72 tiles in an 8-by-9 rectangle. • square tiles • a calculator
There are many ways to arrange 72 tiles into a rectangle. Two more ways are shown below.
4 18
Remembeemember 6 12 • A factor of the The number of tiles is the area of the rectangle, 72. whole number N is Each side length, 8 and 9, 4 and 18, and 6 and 12, is a factor a number you can multiply by another of the area, 72. whole number to The area, 72, is a multiple of each factor: 4, 6, 8, 9, 12, and 18. get N. e.g., 5 is a factor of 10, • What are 4 other numbers of tiles less than 100 that have a lot of since 5 3 2 5 10. possible rectangles? • A multiple of a whole number is the result ______48 ______60 ______80 ______96 of multiplying the whole number by a whole number. e.g., 10 is a multiple of 5, since 5 3 2 5 10. • Any multiplication sentence describes factors and multiples.
167 Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Factors and Multiples, Pathway 3 • Choose 2 of the 4 numbers that you wrote on page 167. List all the factors. Explain or show how you know each is a factor.
number: ______e.g., 60 factors: e.g.,______1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Explain or show how you know each is a factor. e.g., I made these rectangles with 60 tiles and the side lengths are all factors of 60:
number: ______e.g., 80 factors: ______e.g., 1, 80, 2, 40, 4, 20, 5, 16, 8, 10
Explain or show how you know each is a factor. e.g., 80 can be written as 1 x 80, 2 x 40, 4 x 20, 5 x 16, and 8 x 10 so all of those numbers are factors.
• Choose one of the 2 numbers above and then choose 6 of its factors.
number: ______e.g., 60 6 factors: ______e.g., 1, 2, 3, 6, 10, 20 • List 3 multiples of each factor that are greater than 200. Explain or show how you know each is a multiple.
e.g., multiples of 1: 201, 202, 203 multiples of 6: 240, 246, 252 201 x 1 = 201, 202 x 1 = 202, 203 x 1 = 203 240 = 6 x 40, 246 = 6 x 41, 252 = 6 x 42 multiples of 2: 202, 204, 206 multiples of 10: 220, 230; 240 They are even. 220 = 10 x 22, 230 = 10 x 23, 240 = 10 x 24 multiples of 3: 303, 306, 309 multiples of 20: 400, 800, 1200 303 = 3 x 101, 306 = 3 x 102, 309 = 3 x 103 400 = 20 x 20; 800 = 20 x 40, 1200 = 20 x 60
168 Factors and Multiples, Pathway 3 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd. Factors and Multiples Pathway 3 GUIDED
12 students are standing in 3 equal rows of 4. You will need
• square tiles • a calculator
Remembeemember
• A factor of the whole number N is a number you can multiply by another whole number to get N. e.g., 5 is a factor of 10, since 5 3 2 5 10. • A multiple of a whole number is the result of multiplying the whole number by a whole number. • You can use multiplication to describe the way e.g., 10 is a multiple these students are arranged: 3 3 4 5 12. of 5, since 5 3 2 5 10. The factors are the number of groups or rows (3) 3 5 • Any multiplication and the number of students in each row (4). 3 4 12 a Q a sentence describes The multiple is the number of students (12). factors multiple factors and multiples.
Factors • To figure out if a number is a factor of another number, you can create rectangles using equal rows of square tiles. For example, to find factors of 30 you can create rectangles with an area of 30. 5 and 6 are factors of 30 because you can 4 is not a factor of 30 because you can’t make a rectangle of area 30 and the side make a rectangle of area 30 with a side lengths are 5 and 6. length of 4.
4 5
6
169 Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Factors and Multiples, Pathway 3 • You can also divide to figure out factors. For example, you can divide 30 by whole numbers 30 or less. If there is no remainder, the number you are dividing by is a factor of 30. 30 4 3 5 10, so 3 and 10 are factors of 30. 30 4 4 5 7 R2, so 4 and 7 are not factors of 30.
Multiples • You can also use rectangles to figure out multiples of a number. For example, to find multiples of 9, you can make rectangles with side lengths of 9. 36 is a multiple of 9 since you can make a rectangle of area 36 with a side length of 9.
4
9
Notice also that the rectangle shows that 36 is a multiple of 4. • You can also figure out multiples by multiplying. For example, multiply 9 by different whole numbers. 18 and 27 are multiples of 9 because 9 3 2 5 18 and 9 3 3 5 27. Try These 1. What does each rectangle tell you about factors and multiples? a) c)
e.g., 5 and 12 are factors of 60. e.g., 9 and 5 are factors of 45.
60 is a multiple of 5 and 12. 45 is a multiple of 9 and 5. b) d)
e.g., 7 and 4 are factors of 28. e.g., 4 and 8 are factors of 32.
28 is a multiple of 7 and 4. 32 is a multiple of 4 and 8.
170 Factors and Multiples, Pathway 3 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd. 2. Decide if each statement is true or false. Explain your thinking. Remembeemember a) 102 is a multiple of 3. • Another way to figure out factors of a true; e.g., 3 x 34 = 102 number is by using a b) 606 is a multiple of 6. factor tree. You start with the number and true; e.g., 6 x 101 = 606 keep breaking it up into factors. c) 309 is a multiple of 9. e.g., 40 false; e.g., 9 x 34 = 306 and 9 x 35 = 315 and 309 is between 306 and 315. d) 8 is a factor of 512. 4 10 2 2 2 5 true; e.g., 8 x 64 = 512 40 5 2 3 2 3 2 3 5 5 3 3 e) 7 is a factor of 144. 40 4 2 5 40 5 8 3 5 40 5 4 3 10 false; e.g., 7 x 20 = 140 and 7 x 21 = 147 and 144 is in between. The factors of 40 are f) 7 is a factor of 770. 2, 4, 5, 8, and 10.
true; e.g., 7 x 110 = 770
3. a) How does this factor tree show the factors of 45? 45
e.g., It shows 3 x 3 x 5 = 45 and 9 x 5 = 45. It also 9 5 3 3 shows 15 x 3 = 45, if you combine one of the 3s with
the 5. So it shows the factors 3, 5, 9, and 15. b) Use factor trees to list all the factors of 72 and 120. If a factor in your list is not on the tree, tell what factors from the tree can be used to create it.
72 120
2, 3, 4, 6 (2 x 3), 8, 9, 12 (4 x 3), 2, 3, 4, 5, 6 (2 x 3), 8 (2 x 4), 10, 12, 15 (5 x 3), 18 (2 x 9), 24 (8 x 3), 36 (9 x 4), 72 20 (4 x 5), 24 (2 x 12), 30 (3 x 10), 40 (4 x 10), 60 (5 x 12), 120
4. If 8 is a factor of a number, what other numbers have to be factors of the number? How do you know?
1, 2, 4; e.g., You can break each group of 8 into groups of 4, 2, and 1.
171 Copyright © 2012 by Nelson Education Ltd. Leaps and Bounds Factors and Multiples, Pathway 3 5. a) List the first 10 multiples of 4 and the first 10 multiples of 8.
multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40;
multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80
b) What do you notice? e.g., Every other number in the list of
multiples of 4 is in the list of multiples of 8.
6. If 30 is a multiple of a number, how do you know that 60 and 90 must also be multiples of the number?
e.g., if 30 = █ x C, then 60 = 2 x █ x C and 90 = 3 x █ x C
7. How do you know there is a multiple of 7 greater than 1000?
e.g., 7000 is a multiple of 7 (7 x 1000).
8. a) Packages 6 cm high are placed one on top of another in a stack that is almost 50 cm tall. How high is the stack?
e.g., 48 cm 6 x 8 = 48
b) A row of identical books fits perfectly on a shelf that is Encyclopedia Vol.Encyclop 1 Encyclopedia Vol. 2 Encyclopedia Vol. 3 Encyclopedia Vol. 4 Encyclopedia Vol. 5 Encyclopedia Vol. 6 Encyclopedia Vol. 7 Encyclopedia Vol. 8 Encyclopedia Vol. 9 Encyclopedia Vol. 10 Encyclopedia Vol. 11 Encyclopedia Vol. 12 Encyclopedia Vol. 13 Encyclopedia Vol. 14 Encyclopedia Vol.Encyclop 15 96 cm wide. The thickness of each book is a whole Vol edia Vol Vol edia Vol . 1 .1 . 1 1 number of centimetres. How thick might the books be? 5 Show 3 possible solutions.
e.g., 2 cm because 2 x 48 = 96 3 cm because 3 x 32 = 96 4 cm because 4 x24 = 96
9. Do greater numbers always have more factors? Explain your thinking. FYI Factoring whole Not necessarily; e.g., 23 has fewer factors than 22, but bigger numbers is helpful in everyday life in numbers have a greater chance of having more factors. deciding on group sizes and is also helpful in algebra.
172 Factors and Multiples, Pathway 3 Leaps and Bounds Copyright © 2012 by Nelson Education Ltd.