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Counting is a part of everyday life. It happens so often that it almost comes naturally. Think about deciding the amount of maple syrup poured on pancakes or keeping track of the number of snacks left in the pantry. Those are just a few examples for personal use. This lesson will explore situations where counting helps divide things into equal amounts.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

## Rectangular Arrangements

Take a look at the different arrangements of the squares in the applet. Switch between and squares to see the different arrangements for each option. Consider vertical and horizontal arrangements as different arrangements. Now, think about the following questions.
• Consider when there are five squares. How many of the arrangements are rectangles?
• Think about each arrangement as a division. What do the orange squares mean?
• Consider the two previous questions for when there are six squares.

## Counting Is Everywhere

Numbers are used to counting things in everyday life. The most basic numbers for counting are natural and whole numbers.

## Natural Numbers

Natural numbers, also known as counting numbers, are the base numbers for counting things and objects. A natural number contains no fraction, decimal part, or negative sign. A set of natural numbers is represented by the letter and contains the numbers from to infinity.

The dots indicate that the numbers continue to increase by one unit at a time indefinitely. This means that this set contains infinitely many numbers.

## Whole Numbers

Whole numbers represented by the symbol are a set consisting of all natural numbers and

The dots indicate that the set continues to increase indefinitely one step at a time. These are some things to consider about whole numbers and natural numbers.

Natural Numbers Whole Numbers
The smallest natural number is The smallest whole number is
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.
Be aware that natural numbers are sometimes defined similarly to whole numbers.

## Number Line

A number line is a horizontal straight line used to represent numbers visually. Numbers are placed along the line at equal intervals jointly with a vertical segment at each interval. The arrows in the line indicate that the line extends infinitely on the left- and right-hand sides. A number on the left-hand side of the line is always less than a number on the right-hand side.

### Natural Numbers on a Number Line

Natural numbers can be represented in a number line that starts from Then and so on, are marked at equal distances along the line to the right of ### Whole Numbers on a Number Line

A number line can also be used to represent whole numbers. The line begins at and the rest of the whole numbers are drawn along the line to the right. ## Whole Numbers in a Number Line

Find the whole number represented by the point on the number line. ## Relationship Between Division and Multiplication

The product of two or more numbers can be divided evenly by each number in the multiplication expression. This is true every time multiplication is performed. The concepts of factors and multiples are important characteristics to understand when performing such multiplication and division. Consider their definitions.

## Factor

A number is a factor of another number if divides evenly and there is no remainder. This also means that the resulting number is such that the product of and equals

The numbers and are factors of because of the Commutative Property of Multiplication. For this reason, each multiplicand in a multiplication expression is called a factor. A number can have many factors.

Number Factors

## Multiple

Consider when a number is multiplied by an integer. The result is a multiple of the first number. One example shows that is a multiple of This also means that is a factor of Therefore, a number is a multiple of a number if and only if is a factor of

## Giving Equal Amounts of Candy

Dominika loves Halloween. She plans to give candy to her friends at a Halloween party. Her parents give her a bag containing pieces of candy. She wants to divide the candy into equal amounts.

Find the following information to help Dominika have a great Halloween party.

a Write the factors of from least to greatest.
b Dominika decides to give four pieces of candy to some of her friends. How many friends can she give the candy to?
c After giving away all of the candy, Dominika asks her parents for more. She wants a total of pieces of candy. How many bags of pieces should Dominika's parents buy?

### Hint

a Make a list of pairs of whole numbers whose product is Stop the list when the factors begin to repeat.
b Divide by
c List some multiples of until the total is What number must be multiplied by to get

### Solution

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 evenly are needed. This is the same as finding all pairs of whole numbers whose product is These numbers can be found by making a list.
Factors of
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.
b Dominika decides to give four pieces of candy to some of her friends. Divide by to find how many friends she can give the candy to.
This means that Dominika can give pieces of candy to of her friends before she runs out of candy.
c Dominika wants additional pieces of candy. Consider that each bag of candy contains pieces. List the multiples of until appears to find how many bags her parents should buy.
Multiples of

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

## Creating Halloween Decorations

Oh no! Dominika forgot to arrange balloons around her house for the party. She has balloons. She wants to create arrangements of balloons without having any balloons leftover. She also has boxes of spider decorations to decorate the walls.

Help Dominika complete her Halloween decorations!

a Dominika creates the balloon arrangements with the balloons she has. Which of the following statement describes this situation?
b Dominika's little brother claims that there are spider decorations in total. Is this claim true?

### Hint

a Is a factor of
b Check if is a multiple of

### Solution

a Dominika wants to create the balloon arrangements without having any balloons leftover. The total number of balloons must be divided evenly by to accomplish that. That is, must be a factor of The product of and some number is if is a factor of Try to find that number.
There is not a whole number that can be multiplied by to equal This means that is not a factor of Dominika cannot create the balloon arrangements without having any balloons leftover. She could go for balloon arrangements instead, though, because
b Dominika has boxes of spider decorations each. This means that the total number of spider decorations is a multiple of List the multiples of until a number that is greater than or equal to appears.
Notice that is not part of the list of multiples of It is not a multiple of This means that there cannot be only spider decorations like Dominika's little brother claims. The actual total number of decorations can be found by multiplying by
There are actually spider decorations in total. Dominika's house looks really scary now. What an incredible Halloween party this will be!

## Using Factors to Classify Whole Numbers

There are infinitely many whole numbers. Yet, they can be classified into two groups depending on the number of factors they have. Consider, for example, the factors of and

Number Factors Number of Factors
and
and

The number belongs to a set called prime numbers, while belongs to a group called composite numbers.

## Prime Numbers

A prime number is a whole number greater than that is only divisible by and the number itself. In other words, the only factors of a prime number are and itself.

Number Factors Prime?
and
and
and
The set of prime numbers contains infinitely many numbers.

## Composite Numbers

A composite number is a whole number with more than two factors. This means that there is at least another number different than and the number itself that divides this number evenly.

Number Factors Composite?
and
and
and

The set of composite numbers contains infinitely many numbers. These are some facts about prime and composite numbers.

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

## Prime or Composite?

Determine whether the given number is prime or composite. ## What About the Number ?

The number is a unique number because it is neither prime nor composite. Why is this?

### Is Not Prime

The definition of a prime number states that a prime number is a whole number greater than whose factors are and the number itself. Suppose that the restriction about being greater than is removed. List the factors of
It seems that number can be divided by and itself, which is also This fits the definition of a prime number, right? However, recall that every whole number greater than has a unique prime factorization. This means that a whole number can be expressed as the product of prime numbers. Here are some examples.
Number Prime Factorization
Accepting as a prime number means that the prime factorization of a number is no longer unique. Consider the factorization of Any number of ones can be added to any prime factorization according to the Identity Property of Multiplication.
The factor could be included any number of times by this logic. Therefore, any number greater than would have infinite prime factorizations. The restriction about prime numbers being greater than is necessary so cases like this do not happen.

### Is Not Composite

A composite number is defined as having more than two factors. This means that cannot be composite because its only factor is itself.

### Conclusions

Now that it is known that is neither prime nor composite let's summarize these facts.

is not a prime number.
is not a composite number.
The only factor of is itself.

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