Divisibility Rules 1 - Nelson

OPEN-ENDEDPathway 1

You will need• base ten blocks

(optional)• a calculator

Divisibility Rules

I wonder if 72 is

divisible by 9.

We know that 72 is divisible by 2, 3, 6, and 9 because you can make groups of 2, 3, 6, or 9 and have nothing left over.

For example, if you model 72 with blocks, you can make 8 groups of 9 and there are none left over.

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.

• If a number is divisible by another number, when you divide them the answer is a whole number. e.g., 72 4 6 5 12, so 72 is divisible by 6 (and by 12).

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Copyright © 2012 by Nelson Education Ltd.156

Leaps and BoundsDivisibility Rules, Pathway 1

• 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



• Choose 2 of the rules you wrote above and explain why each makes sense.


Leaps and Bounds Divisibility Rules, Pathway 1

If the last digit of a number is even, the number is divisible by 2.

If a number is divisible by both 2 and 3, then it is also divisible by 6.

If a number is divisible by both 2 and 3, then it is also divisible by 6.

If the sum of the digits of a number is divisible by 9, the number is divisible by 9.

e.g., The test for 10: If a number can be grouped in 10s, when you model it with base ten blocks there would be no ones, so 0 would be in the ones place.

If the last digit of a number is 0 or 5, the number is divisible by 5.

If the last digit of a number is 0, the number is divisible by 10.

e.g., The test for 6: A number that can be grouped in 6s can also be arranged in pairs (each group of 6 is 3 pairs) 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.

GUIDEDPathway 1

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, 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 divisible by 3.

Divisibility by 3 Rule

If the sum of the digits of a number is divisible by 3, then the number is divisible by 3.

You can look at the pattern in the sum of the digits for multiples of 3 from 0 to 57 to see why this rule works:

Multiples of 3

Multiple of 3 0 3 6 9 12 15 18 21 24 27

Sum of digits 0 3 6 9 3 6 9 3 6 9

Multiple of 3 30 33 36 39 42 45 48 51 54 57

Sum of digits 3 6 9 12 6 9 12 6 9 12

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.

Divisibility Rules

• If a number is divisible by another number, the result will be a whole number when you divide. e.g., 72 4 6 5 12, so 72 is divisible by 6 (and by 12).

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Divisibility Rules, Pathway 1 Leaps and Bounds

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 RuleIf the sum of the digits of a number is divisible by 9, the number is divisible by 9.

Can 428 cookies be in packages of 9?

4 1 2 1 8 5 14, and 14 is not a multiple of 9, so 428 cookies cannot be packaged in 9s.

Divisibility by 2 RuleIf the ones digit of a number is even, the number is divisible by 2.


The digit 8 is even, so 428 cookies can be packaged in 2s.

Divisibility by 5 and 10 RuleIf the ones digit of a number is 5 or 0, the number is divisible by 5.If the ones digit of a number is 0, the number is divisible by 10.

Can 428 cookies be in packages of 5 or 10?

The ones digit is 8, not 5 or 0, so 428 cookies cannot be packaged in 5s or 10s.

Divisibility by 6 RuleIf the ones digit of a number is even and the sum of its digits is divisible by 3, then the number is divisible by 6.


8 is even, but 4 1 2 1 8 5 14, which is not divisible by 3, so 428 cookies cannot 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?

b) Which are divisible by 3?

c) Which are divisible by 5?

d) Which are divisible by 6?

e) Which are divisible by 9?

f) Which are divisible by 10?

2. Every answer for Question 1e) was also an answer for 1b), but not the other way around. Explain why.

615 387

258

610 490

369

612 429


Divisibility Rules, Pathway 1Leaps and Bounds

258, 612, 490, 610

258, 429, 387, 612, 369, 615

490, 615, 610

258, 612

387, 612, 369

490, 610

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.

3. Write digits in the blanks to make numbers that will make the statements true. Complete each statement in 2 ways.

a) 5 is divisible by 9.

b) 26 is divisible by 6.

c) 3 4 is divisible by 2.

d) 69 2 is divisible by 3 and 10.

5 is divisible by 9.

26 is divisible by 6.

3 4 is divisible by 5.

69 2 is divisible by 3 and 10.

4. a) Complete this chart for multiples of 9.

Multiple of 9 90 99 108 117 126 135 144 153 162 171 180 189

Sum of digits

b) Describe the patterns in the multiples of 9.

c) What do you notice about the sums of the digits?

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.

5. Choose a divisibility rule for 2, 5, or 10. Explain why it makes sense.


Divisibility Rules, Pathway 1 Leaps and Bounds

4 4

4

9 9

2 2

2 2

1

1

0

00

0 0

9 9 9 9 9 9 9 1818

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.

e.g., The sum of the digits is usually 9 but sometimes 18.

e.g., The sum of the digits is 9 or 18, and both are divisible by 9.

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.

999

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.”

“I think a number is divisible by 6 if the sum of its digits is 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 is divisible by 3, then the number is divisible by 6.

Do you agree with what Tyler is saying? Explain your thinking.

8. Think about Tyler’s divisibility by 6 rule. How could you use a similar idea to create a divisibility by 15 rule?

9. Why do you think divisibility rules were more useful in the past than they are now?

If a number is

divisible by 6, it is

divisible by both

2 and 3.

Knowing about divisibility rules can make it easier to determine if a number is prime or composite.

FYI


Divisibility Rules, Pathway 1Leaps and Bounds

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.

e.g., The sum of the digits of 51 is 6 and 51 is not divisible by 6.

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.

e.g., Combine the tests for 3 and 5.

e.g., Almost everyone has a calculator now.

OPEN-ENDEDPathway 2

Part A

Some numbers of tiles can be arranged in different rectangles. Some numbers of tiles can be arranged in only 1 rectangle.

I made 3 different rectangles

with 12 tiles but only

1 rectangle with 11 tiles.

If tiles can be arranged in 2 or more different rectangles, that number of tiles is a composite number. For example, 12 is a composite number.

If only 1 rectangle is possible, the number is a prime number. For example, 11 is a prime number.

The number 1 is neither prime nor composite.

• Figure out which numbers from 2 to 50 are prime and which are composite. List them below.

prime: composite:

• Is the sum of 2 prime numbers usually a prime number? Explain.

• Can the product of 2 prime numbers be a prime number? Explain.

Prime Numbers and Perfect Squares

You will need• square tiles• a calculator

compositea whole number with more than 2 factors e.g., 6 is composite because 6 5 2 3 3 and 6 5 1 3 6.

primea whole number with exactly 2 factors—itself and 1e.g., 7 is prime because its only factors are 7 and 1 (7 5 7 3 1).


Leaps and BoundsPrime Numbers and Perfect Squares, Pathway 2

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

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.

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

Part B

If a number of tiles can be arranged in a square, that number of tiles is called a perfect square.

The side length of the square is the square root of the number of tiles.

For example, 4 is a perfect square, and 2 is the square root of 4.

2 2 4

2

• Complete the charts, using numbers from 1 to 200 that are perfect squares. Write the square root for each number.

Squares and Square Roots

Perfect squares

Square roots

Perfect squares

Square roots

• Can the sum of 2 perfect squares be a perfect square? If so, do you think this is always true? Explain, using examples.

• Can the product of 2 perfect squares also be a perfect square? Explain, using examples.

perfect squarea number that is the product of a whole number multiplied by itself e.g., 9 is a perfect square because 3 3 3 5 9.

square roota number that you can multiply by itself to get a given number e.g., 4 is the square root of 16 because 4 3 4 5 16.


Leaps and Bounds Prime Numbers and Perfect Squares, Pathway 2

1

1

4 9 16 25 36 49

2 3 4 5 6 7

64 81 100 121 144 169 196

8 9 10 11 12 13 14

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.

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).

GUIDEDPathway 2

Some numbers are given special names because of the way they can be written as products.

Prime and Composite Numbers

• 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.

• 3 square tiles can be arranged in only 1 rectangle — a 1-by-3 rectangle. That means you can write 3 as 1 3 3, and the factors of 3 are 1 and 3.

1 3

Since 3 can be arranged in only 1 rectangle and it has exactly 2 factors, 3 is a prime number.

• The number 1 is neither a prime nor a composite because it has only 1 factor.

Perfect Squares and Square Roots

• 4 tiles can be arranged in a square with a side length of 2. 9 tiles can be arranged in a square with a side length of 3.

2 2 4 3 3 9

2

3

That means the numbers 4 and 9 are perfect squares and their side lengths are called square roots.

2 is the square root of the perfect square 4. 3 is the square root of the perfect square 9.

Prime Numbers and Perfect Squares


compositea whole number with more than 2 factors e.g., 6 is composite because 6 5 2 3 3 and 6 5 1 3 6.

primea whole number with exactly 2 factors —itself and 1e.g., 7 is prime because its only factors are 7 and 1 (7 5 7 3 1).

perfect squarea number that is the product of a whole number multiplied by itself e.g., 9 is a perfect square because 3 3 3 5 9.

square roota number that you can multiply by itself to get a given number e.g., 4 is the square root of 16 because 4 3 4 5 16.



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?

3. Can a prime number be even? Explain your thinking.

4. Choose 2 composite numbers. Write each as a product of prime numbers. For example, 24 5 2 3 2 3 2 3 3.

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.

b) Would you divide 143 by any numbers greater than 100 to see if they are factors? Explain your thinking.

c) Would you divide 143 by 11 to see if it is a factor? Explain.


Leaps and Bounds Prime Numbers and Perfect Squares, Pathway 2

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.

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).

e.g., 40 = 2 x 2 x 2 x 5

65 = 5 x 13

No; e.g., If 3 is not a factor, 9 can't be.

No; e.g., 100 x 2 is already 200 which is too big.

Yes; e.g., 11 is a factor because 11 x 13 = 143, so 143 is not prime.

6. Can the sum of 2 prime numbers be prime? Explain.

7. Circle each perfect square and write its square root.

49 81 90 121 141 169

8. Do you agree with what Karma says at the right? Explain your thinking.

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.

b) Is the sum of 2 perfect squares always a perfect square? Explain your thinking using an example.

c) Can the product of 2 perfect squares be a perfect square? Explain your thinking using an example.

10. a) Why do you think there are a lot fewer prime numbers than composite numbers?

b) Why do you think there are a lot fewer perfect squares than other whole numbers?

There are no

perfect squares

between 225

and 256.

Knowing about prime numbers will help you know when a number has been factored as much as it can be.

FYI



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, 9, 11, 13

Yes, e.g., 225 = 15 x 15 and 256 = 16 x 16 and there are no whole

numbers between 15 and 16.

e.g., 36 + 64 = 100

No, e.g., 9 + 25 = 34, which is not a perfect square.

Yes, e.g., 9 x 16 = 144 and 144 is 12 x 12.

e.g., Every second number is even and a lot of the odd numbers are multiples of 3 or 5, so a lot of numbers are composite.

e.g., Square numbers are 1, 4, 9, 16, and so on and they get farther apart with many other whole numbers between them.

OPEN-ENDEDPathway 3

I arranged 72 tiles in

an 8-by-9 rectangle.

There are many ways to arrange 72 tiles into a rectangle. Two more ways are shown below.

4 18

6 12

The number of tiles is the area of the rectangle, 72.

Each side length, 8 and 9, 4 and 18, and 6 and 12, is a factor of the area, 72.

The area, 72, is a multiple of each factor: 4, 6, 8, 9, 12, and 18.

• What are 4 other numbers of tiles less than 100 that have a lot of possible rectangles?

________ ________ ________ ________

Factors and Multiples


• 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. e.g., 10 is a multiple of 5, since 5 3 2 5 10.

• Any multiplication sentence describes factors and multiples.

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Leaps and Bounds Factors and Multiples, Pathway 3

48 60 80 96

• 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: ________ factors: ________________________________

Explain or show how you know each is a factor.

number: ________ factors: ________________________________

Explain or show how you know each is a factor.

• Choose one of the 2 numbers above and then choose 6 of its factors.

number: ________ 6 factors: _______________________________

• List 3 multiples of each factor that are greater than 200. Explain or show how you know each is a multiple.


Leaps and BoundsFactors and Multiples, Pathway 3

e.g., 60 e.g., 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

e.g., I made these rectangles with 60 tiles and the side lengths are all factors of 60:

e.g., 80 e.g., 1, 80, 2, 40, 4, 20, 5, 16, 8, 10

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.

e.g., 60 e.g., 1, 2, 3, 6, 10, 20

e.g., multiples of 1: 201, 202, 203 201 x 1 = 201, 202 x 1 = 202, 203 x 1 = 203 multiples of 2: 202, 204, 206 They are even. multiples of 3: 303, 306, 309 303 = 3 x 101, 306 = 3 x 102, 309 = 3 x 103

multiples of 6: 240, 246, 252 240 = 6 x 40, 246 = 6 x 41, 252 = 6 x 42 multiples of 10: 220, 230; 240 220 = 10 x 22, 230 = 10 x 23, 240 = 10 x 24 multiples of 20: 400, 800, 1200 400 = 20 x 20; 800 = 20 x 40, 1200 = 20 x 60

GUIDED


Pathway 3

12 students are standing in 3 equal rows of 4.

• You can use multiplication to describe the way these students are arranged: 3 3 4 5 12.

The factors are the number of groups or rows (3) and the number of students in each row (4).

The multiple is the number of students (12).

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 make a rectangle of area 30 and the side lengths are 5 and 6.

5

6

4 is not a factor of 30 because you can’t make a rectangle of area 30 with a side length of 4.

4

3 3 4 5 12a Q afactors multiple

Factors and Multiples

• 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. e.g., 10 is a multiple of 5, since 5 3 2 5 10.

• Any multiplication sentence describes factors and multiples.

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• 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)

b)

d)



e.g., 5 and 12 are factors of 60.

60 is a multiple of 5 and 12.

e.g., 9 and 5 are factors of 45.

45 is a multiple of 9 and 5.

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.

2. Decide if each statement is true or false. Explain your thinking.

a) 102 is a multiple of 3.

b) 606 is a multiple of 6.

c) 309 is a multiple of 9.

d) 8 is a factor of 512.

e) 7 is a factor of 144.

f) 7 is a factor of 770.

3. a) How does this factor tree show the factors of 45?

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

4. If 8 is a factor of a number, what other numbers have to be factors of the number? How do you know?

45

9 5

3 3

• Another way to figure out factors of a number is by using a factor tree. You start with the number and keep breaking it up into factors. e.g.,

40

4 10

2 2 2 5 40 5 2 3 2 3 2 3 5

40 5 4 3 2 3 540 5 8 3 540 5 4 3 10

The factors of 40 are 2, 4, 5, 8, and 10.

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true; e.g., 3 x 34 = 102

true; e.g., 6 x 101 = 606

false; e.g., 9 x 34 = 306 and 9 x 35 = 315 and 309 is between 306 and 315.

true; e.g., 8 x 64 = 512

false; e.g., 7 x 20 = 140 and 7 x 21 = 147 and 144 is in between.

true; e.g., 7 x 110 = 770

e.g., It shows 3 x 3 x 5 = 45 and 9 x 5 = 45. It also

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.

2, 3, 4, 6 (2 x 3), 8, 9, 12 (4 x 3), 18 (2 x 9), 24 (8 x 3), 36 (9 x 4), 72

2, 3, 4, 5, 6 (2 x 3), 8 (2 x 4), 10, 12, 15 (5 x 3), 20 (4 x 5), 24 (2 x 12), 30 (3 x 10), 40 (4 x 10), 60 (5 x 12), 120

1, 2, 4; e.g., You can break each group of 8 into groups of 4, 2, and 1.

5. a) List the first 10 multiples of 4 and the first 10 multiples of 8.

multiples of 4:

multiples of 8:

b) What do you notice?

6. If 30 is a multiple of a number, how do you know that 60 and 90 must also be multiples of the number?

7. How do you know there is a multiple of 7 greater than 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?

b) A row of identical books fits perfectly on a shelf that is 96 cm wide. The thickness of each book is a whole number of centimetres. How thick might the books be? Show 3 possible solutions.

9. Do greater numbers always have more factors? Explain your thinking.

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Factoring whole numbers is helpful in everyday life in deciding on group sizes and is also helpful in algebra.

FYI



4, 8, 12, 16, 20, 24, 28, 32, 36, 40;

8, 16, 24, 32, 40, 48, 56, 64, 72, 80

multiples of 4 is in the list of multiples of 8.

e.g., 7000 is a multiple of 7 (7 x 1000).

e.g., if 30 = █ x C, then 60 = 2 x █ x C and 90 = 3 x █ x C

e.g., 48 cm 6 x 8 = 48

e.g., 2 cm because 2 x 48 = 96 3 cm because 3 x 32 = 96 4 cm because 4 x24 = 96

Not necessarily; e.g., 23 has fewer factors than 22, but bigger

numbers have a greater chance of having more factors.

e.g., Every other number in the list of

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Divisibility Rules 1 - Nelson