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

a A factor of a specific number is another number that divides that number evenly, without a remainder. In this case, the numbers that divide

20

evenly are needed. This is the same as finding all pairs of whole numbers whose product is

20 .

These numbers can be found by making a list.

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

The list stops when the factors begin to repeat but in the opposite order. In this case, the last row contains the same factors as in the row before it but in reverse order. Now, list all the factors from least to greatest.

Factors of 20 : 1, 2, 4, 5, 10, 20

b Dominika decides to give four pieces of candy to some of her friends. Divide

20

4

to find how many friends she can give the candy to.

\frac{2 0}{4} = 5 friends

This means that Dominika can give

4

pieces of candy to

5

of her friends before she runs out of candy.

c Dominika wants

120

additional pieces of candy. Consider that each bag of candy contains

20

pieces. List the multiples of

20

until

120

appears to find how many bags her parents should buy.

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

The sixth multiple of $20$ is $120 .$ This means that Dominika's parents should buy $6$ bags of candy. That is because $20 \cdot 6 = 120 .$ Great! The candy is ready and Dominika can now have fun at the Halloween party.

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

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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

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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

$1$ Is Not Prime

$1$ Is Not Composite