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

LaShay wants to determine which of the following numbers are prime numbers.

82, 89, 107, 118, 131

Help LaShay determine the prime numbers in this set of numbers.

We are asked to identify which of the numbers in the given set are prime numbers. 82, 89, 107, 118, 131 Let's start by recalling the definition of prime numbers.

A prime number is a whole number greater than 1 that is only divisible by 1 and the number itself.

We have a great process to check if a number is a prime number. Begin by trying to divide the number by 2. If it is evenly divisible by 2, then it cannot be a prime number unless the number is 2. If the number is not evenly divisible by 2, then we will try to divide it by 3, the next number. We will find and list the factors of the number 82 this way. Factors of82: 1, 2, 41, 82 We found that 82 has four factors. It is not a prime number. Now, let's look at the factors of other numbers in the given list.

Number	Factors	Is It Prime?
82	1, 2, 41, and 82	No
89	1 and 89	Yes
107	1 and 107	Yes
118	1, 2, 59, and 118	No
131	1 and 131	Yes

We found that 89, 107, and 131 are the prime numbers in the list.

We can use the applet to determine the factors of the numbers in the list and check our answer.

Mark was stuck on his homework for a while. He is struggling determining which numbers in the set are composite.

29, 202, 169, 79, 97

Select the composite numbers to help Mark.

We are asked to determine which of the numbers in the given set are composite numbers. 29,202,169,79,97 Let's begin by recalling the definition of composite numbers.

A composite number is a whole number with more than two factors.

There is a tried and true method to determine if a number is composite. We can begin by checking if two is a factor of that number. Next, we can check if 3 is a factor. We can find all of its factors this way. Now, let's find the factors of 29. Factors of29: 1,29 The number 29 has only two factors. Therefore, it is not a composite number. Now, let's look at the factors of the other numbers in the list.

Number	Factors	Is It Composite?
29	1, 29	No
202	1, 2, 101, 202	Yes
169	1, 13, 169	Yes
79	1, 79	No
97	1, 97	No

We can use the applet to determine the factors of the numbers in the list and check our answer.

Select the number of factors each of the following numbers has.

543

342

832

We will first list all the factors of 543. We will then count the numbers in the list to find the number of factors 543 has. We can create this list by checking if the number 1 is a factor. Then we will check if 2 is a factor. We will find and list the factors of the number 543 this way. Factors of543 [-1.1em] 1, 3, 181, 543 We have 4 different numbers in the list. This means that 543 has 4 factors.

We can find the factors of 342 by following a similar process. Let's create a list containing the factors of 342. Factors of342 [-1.1em] 1, 2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342 We can now count how many numbers we have on the list. We can see that there are 12 different numbers on this list. So, 342 has 12 factors.

Finally, let's list the factors of 832. Factors of832 [-1.1em] 1, 2, 3, 4, 8, 13, 16, 26 32, 52, 64, 104, 208 , 416, 832 There are 14 numbers in the list we created. This means that 832 has 14 factors.

We can use the applet to determine the factors of the numbers.

Dylan followed this process to determine whether the number $37$ is prime or composite.

Which step has the mistake?

We will go through the process to determine if 37 is prime or composite. We can then compare each of our steps with Dylan's step. That will help us to identify the step where Dylan made a mistake. Let's first find the factors of 37. Factors of37: 1,37 The only factors of 37 are 1 and itself. We can see that this matches with Dylan's first step. Next, we can state that the number has only two factors. Factors of37: 1,37 Number of Factors:2 Dylan also stated the number of factors correctly. Now, recall that a prime number is a number with only two factors. Conversely, a composite number has more than two factors. This means that 37 is a prime number because it has only two factors. 37Is Prime or Composite? Prime Dylan stated that 37 is a composite number. We found that this is not true. This means that Step III has a mistake. Let's correct Dylan's process.

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

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

Natural, Whole, and Prime Numbers

Recommended exercises

$1$ Is Not Prime

$1$ Is Not Composite