Explore Natural Numbers, Whole Numbers, Prime Numbers, and Factors

Natural Numbers	Whole Numbers
$N = {1, 2, 3, 4, 5, \dots}$	$W = {0, 1, 2, 3 \dots}$
The smallest natural number is $1 .$	The smallest whole number is $0 .$
All natural numbers are whole numbers	Not all whole numbers are natural numbers.
Example: Marky's favorite toys are three car figurines, two puzzles, and a guitar.	Example: The school principal has one notebook and ten pencils, yet he has zero erasers.

Natural Numbers

Whole Numbers

N = {1, 2, 3, 4, 5, \dots}

W = {0, 1, 2, 3 \dots}

The smallest natural number is

1 .

The smallest whole number is

0 .

All natural numbers are whole numbers

Not all whole numbers are natural numbers.

Example: Marky's favorite toys are three car figurines, two puzzles, and a guitar.

Example: The school principal has one notebook and ten pencils, yet he has zero erasers.

Number	Factors
$48$	$1,$ $2,$ $3,$ $4,$ $6,$ $8,$ $12,$ $16,$ $24,$ $48$
$21$	$1,$ $3,$ $7,$ $21$
$9$	$1,$ $3,$ $9$

Number

Factors

48

1,

2,

3,

4,

6,

8,

12,

16,

24,

48

21

1,

3,

7,

21

9

1,

3,

9

Factors of $20$
$20 = 1 \cdot 20$
$20 = 2 \cdot 10$
$20 = 4 \cdot 5$
$20 = 5 \cdot 4$

Factors of

20

20 = 1 \cdot 20

20 = 2 \cdot 10

20 = 4 \cdot 5

20 = 5 \cdot 4

Multiples of $20$
$20 \cdot 1 = 20$
$20 \cdot 2 = 40$
$20 \cdot 3 = 60$
$20 \cdot 4 = 80$
$20 \cdot 5 = 100$
$20 \cdot 6 = 120$

Multiples of

20

20 \cdot 1 = 20

20 \cdot 2 = 40

20 \cdot 3 = 60

20 \cdot 4 = 80

20 \cdot 5 = 100

20 \cdot 6 = 120

Multiples of $5$	$5,$ $10,$ $15,$ $20,$ $25,$ $30,$ $35,$ $40,$ $45,$ $50,$ $55,$ $60$

Number	Factors	Number of Factors
$11$	$1$ and $11$	$2$
$9$	$1,$ $3,$ and $9$	$3$

Number

Factors

Number of Factors

11

1

and

11

2

9

1,

3,

and

9

3

Number	Factors	Prime?
$5$	$1$ and $5$	$✓$
$6$	$1,$ $2,$ $3,$ and $6$	$\times$
$7$	$1$ and $7$	$✓$

Number

Factors

Prime?

5

1

and

5

✓

6

1,

2,

3,

and

6

\times

7

1

and

7

✓

Number	Factors	Composite?
$4$	$1,$ $2,$ and $4$	$✓$
$6$	$1,$ $2,$ $3,$ and $6$	$✓$
$7$	$1$ and $7$	$\times$

Number

Factors

Composite?

4

1,

2,

and

4

✓

6

1,

2,

3,

and

6

✓

7

1

and

7

\times

Prime	Composite
A prime number has only two factors.	A composite number has more than two factors.
The only even prime number is $2 .$	Every even number greater than $2$ is composite.
Any whole number greater than $1$ is either prime or composite.
The number $1$ is neither prime nor composite.

Prime

Composite

A prime number has only two factors.

A composite number has more than two factors.

The only even prime number is

2 .

Every even number greater than

2

is composite.

Any whole number greater than

1

is either prime or composite.

The number

1

is neither prime nor composite.

Number	Prime Factorization
$2$	$2 = 2$
$3$	$3 = 3$
$4$	$4 = 2 \cdot 2$
$5$	$5 = 5$
$6$	$6 = 2 \cdot 3$
$12$	$6 = 2 \cdot 2 \cdot 3$

Number

Prime Factorization

2

2 = 2

3

3 = 3

4

4 = 2 \cdot 2

5

5 = 5

6

6 = 2 \cdot 3

12

6 = 2 \cdot 2 \cdot 3

The number line has four points on it that represent whole numbers.

Match each point with the whole number that it represents.

We want to determine which whole numbers the given points represent. We can find this information by using the numbers shown on the number line. Consider 0, 5, 10, or 15 as reference points.

Here are a few tips to help understand how to use reference points. First, consider when the reference point is to the left of the unknown point. Then consider when it is to the right of the unknown point.

Reference point is left of the unknown point: Move from the reference point toward the right to the unknown point. Add 1 for each tick.

Reference point is right of the unknown point: Move from the reference point toward the left to the unknown point. Subtract 1 for each tick.

Now let's try this method to find the value of each of the given points A, B, C, and D.

Point A

We are identifying what the value of point A is on the given number line.

We see that 0 is the point closer to A. Let's use this as our reference number and count how many ticks until we reach the value of A.

We move two units to the right of the number 0 to get to point A. That means we add 2 to 0 to get our answer. We found that A represents 2.

Point B

We will follow a similar process to find the number represented by point B.

Our reference number here is 5. We move 1 unit to the right of the number 5. That means we add 1 to 5 to get our answer. We found that B represents 6.

Point C

Let's now look at point C.

Our reference number here is 10. We move 1 unit to the left of the number 10. That means we subtract 1 from 10 to get our answer. We found that C represents 9.

Point D

Lastly, let's look at point D.

We have treated 15 as our reference point. Point D is two units to the left of 15. That means we subtract 2 from 15 to get our answer. We found that D represents 13.

Find the factors for each number. Write the factors from least to greatest.

27

92

We want to determine the factors of the number 27. This means all whole numbers that divide 27 evenly. This is equivalent to finding pairs of numbers whose product is 27. Each number in these pairs of numbers will be a factor of 27. Let's make a list to find these factors.

Factors of 27
27= 1* 27
27= 3* 9
27= 9* 3

We can see that the factors in the third row are the same as the ones in the second but in a different order. Then, we stop the list. We can now list the factors of 27 from the least to the greatest. Factors of27: 1,3,9,27

We will follow a similar process to find the factors of 92.

Factors of 92
92= 1* 92
92= 2* 46
92= 4* 23
92= 23* 4

The factors begin to repeat in the fourth row, so we stop the list. Now that we have the factors of 92, let's list them from the least to the greatest. Factors of92: 1,2,4,23,46,92

Find the first six multiples of each number. List them from the least to the greatest.

7

71

We can find a multiple of a given number by multiplying it by a whole number. In this case, we want the first six multiples of 7. This means we need to multiply 7 by 1, 2, 3, 4, 5, and 6 one by one. Let's do it!

First Six Multiples of 7
7* 1= 7
7* 2= 14
7* 3= 21
7* 4= 28
7* 5= 35
7* 6= 42

Let's list them from the least to the greatest once we have the first six multiples of 7. Multiples of7 7,14,21,28,35,42

We can find the multiples of 71 in a similar fashion. We will multiply 71 by each number from 1 to 6 one by one.

First Six Multiples of 71
71* 1= 71
71* 2= 142
71* 3= 213
71* 4= 284
71* 5= 355
71* 6= 426

These products are the first six multiples of 71. We can now list them from the least to the greatest. Multiples of71 71,142,213,284, 355,426

Zain wonders if all of the following numbers are factors of

75 .

1, 3, 4, 7, 25, 75

Their friend points out that some of them are not. Identify which ones are not.

We want to find which numbers in Zain's list are not factors of 75. Zain's List 1, 3, 4, 7, 25, 75 We will test each of the numbers in the list one at a time. We will then see if there is a whole number such that the number in the list multiplied by that whole number is 75. The number in the list is not a factor of 75 if there is not such whole number. Let's start with the number 25. 25* 1=25 * 25* 2=50 * 25* 3= 75 ✓ We found that 25 times 3 is 75. This means that 25 and 3 are factors of 75. Let's cross these two numbers off the list because we only want the numbers that are not factors of 75. Zain's List 1, 3, 4, 7, 25, 75 We will follow a similar process to test the remaining numbers. Now, let's test the number 4. 4* 17=68 * 4* 18=72 * 4* 19=76 * 4* 20=80 * The numbers 72 and 76 are close to 75, but not quite. The product goes up if we continue multiplying by the following whole numbers. What does this mean? Any whole number in which 4 times that whole number is 75 should be between 18 and 19. However, no whole number between two consecutive whole numbers exists.

This means that there is not whole number that 4 times that whole number is 75. So, 4 is not a factor of 75. Now, let's see what happens with the next number, 1. 1* 75= 75 ✓ These two numbers are factors of 75. We will discard them from the list. Zain's List 1, 3, 4, 7, 25, 75 Finally, let's see what happens with the number 7. 7* 9=63 * 7* 10=70 * 7* 11=77 * 7* 12=84 * We found that 70 and 77 are close to 75. However, 7 is not a factor of 75 because there is no whole number between 10 and 11. It is safe to say that 7 is not a factor of 75. Zain's List 1, 3, 4, 7, 25, 75 This means that 4 and 7 are not factors of 75.

Emily is quizzing her friend and creates a list with some multiples of

14 .

30, 98, 142, 196, 224

Some of the numbers are not multiples of

14 .

Identify which ones.

We are asked to find the numbers in the list that are not multiples of 14. Emily's List 30, 98, 142, 196, 224 There are methods to identify if a number is multiple of a given number. One way is to find a whole number that times the given number results in the number we are testing. The number is not a multiple of the given number if that whole number does not exist. We will test the numbers in Emily's list to see if they meet this standard.

Checking For Whole Numbers That Meet Our Standard

Let's begin with the number 30 from the list. Then multiply some whole numbers like 1, 2, and 3 by our given multiple of 14 to try and get 30. 1* 14=14 * 2* 14=28 * 3* 14=42 * 4* 14=56 * The numbers 28 and 42 are multiples of 14. They are close to 30, but not quite. The product goes up if we continue multiplying by the following whole numbers. If there is any whole number that times 14 is 30, it should be between 2 and 3. However, consider that no whole number between two consecutive whole numbers exists.

This means that we cannot find a whole number that times 14 is 30, regardless of how hard we try. We can safely say that 30 is not a multiple of 14. Note it on the list. Emily's List 30, 98, 142, 196, 224 Now, let's analyze the number 98. 5* 14=70 * 6* 14=84 * 7* 14= 98 ✓ If we multiply 7 by 14, we get 98. This means that 98 is a multiple of 14. We can discard it from the list because we are looking for numbers that are not multiples of 14. Emily's List 30, 98, 142, 196, 224 The next number on the list is 142. This is close to 140. We know that 10 times 14 equals 140. Next, try 11. The product of 11 and 14 is 154. Well, 142 is between the two results. 10* 14=140 * 11* 14=154 * There is no whole number between 10 and 11 that meets our standard. The same concept was found earlier in this exercise. It is safe to say that 142 is not a multiple of 14. Emily's List 30, 98, 142, 196, 224 Next, we will test the last two numbers in the list, 196 and 224. We can get those numbers by multiplying 14 by 14 and 16 by 14. 14* 14= 196 ✓ 16* 14= 224 ✓ These two numbers are multiples of 14. Let's cross them off the list. Emily's List 30, 98, 142, 196, 224 The two remaining numbers are 30 and 142. They are our champions. They are not multiples of 14.

Natural, Whole, and Prime Numbers

Catch-Up and Review

Rectangular Arrangements

Counting Is Everywhere

Natural Numbers

Whole Numbers

Number Line

Natural Numbers on a Number Line

Whole Numbers on a Number Line

Whole Numbers in a Number Line

Relationship Between Division and Multiplication

Factor

Multiple

Giving Equal Amounts of Candy

Hint

Solution

Creating Halloween Decorations

Hint

Solution

Using Factors to Classify Whole Numbers

Prime Numbers

Composite Numbers

Prime or Composite?

What About the Number $1$ ?

$1$ Is Not Prime

$1$ Is Not Composite

Conclusions

Point A

Point B

Point C

Point D

Checking For Whole Numbers That Meet Our Standard

Natural, Whole, and Prime Numbers

Recommended exercises

	14 Theory slides
	12 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Natural, Whole, and Prime Numbers

Catch-Up and Review

Natural Numbers on a Number Line

Whole Numbers on a Number Line

Hint

Solution

Hint

Solution

1 Is Not Prime

1 Is Not Composite

Conclusions

Point A

Point B

Point C

Point D

Checking For Whole Numbers That Meet Our Standard

Natural, Whole, and Prime Numbers

Recommended exercises

$1$ Is Not Prime

$1$ Is Not Composite