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# Properties of Division

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In this section, we review the basic operation of division with whole numbers. We will discuss a few fundamental properties of division, learn how to identify the parts of a division problem, and learn how to divide single-digit whole numbers when remainders are involved. When we think about a division problem, it has three parts:

Dividend - this is our number that is divided up or split up into equal groups. In a horizontal division problem, it is the leftmost number

Divisor - the amount we are dividing by. In a horizontal division problem, this number follows the division symbol "÷"

Quotient - the result of the division operation

Example 1: Label the parts of 40 ÷ 8 = 5

40 - Dividend

8 - Divisor

5 - Quotient

Next, we discuss a few properties of division:

327 ÷ 1 = 327, dividing a number by 1 leaves the number unchanged

327 ÷ 0 is undefined, division by zero is undefined

327 ÷ 327 = 1, dividing a non-zero number by itself results in 1

Lastly, we cover division with remainders. Not all division works out nicely, sometimes it can be messy. What happens if we see a problem such as:

11 ÷ 2

To visualize the answer, we can count out how many groups of 2 can be subtracted away from 11:

1) 11 - 2 = 9

2) 9 - 2 = 7

3) 7 - 2 = 5

4) 5 - 2 = 3

5) 3 - 2 = 1

After subtracting 5 groups of 2 away from 11, we are left with the number 1. We don't have enough to subtract away another group of 2, so we say that 1 is the leftover amount or remainder from the division.

11 ÷ 2 = 5 Remainder 1

Dividend - this is our number that is divided up or split up into equal groups. In a horizontal division problem, it is the leftmost number

Divisor - the amount we are dividing by. In a horizontal division problem, this number follows the division symbol "÷"

Quotient - the result of the division operation

Example 1: Label the parts of 40 ÷ 8 = 5

40 - Dividend

8 - Divisor

5 - Quotient

Next, we discuss a few properties of division:

- Division by zero is undefined
- Dividing a number by 1 leaves the number unchanged
- Dividing a non-zero number by itself results in the number 1
- Dividing zero by any non-zero number results in the number 0

327 ÷ 1 = 327, dividing a number by 1 leaves the number unchanged

327 ÷ 0 is undefined, division by zero is undefined

327 ÷ 327 = 1, dividing a non-zero number by itself results in 1

Lastly, we cover division with remainders. Not all division works out nicely, sometimes it can be messy. What happens if we see a problem such as:

11 ÷ 2

To visualize the answer, we can count out how many groups of 2 can be subtracted away from 11:

1) 11 - 2 = 9

2) 9 - 2 = 7

3) 7 - 2 = 5

4) 5 - 2 = 3

5) 3 - 2 = 1

After subtracting 5 groups of 2 away from 11, we are left with the number 1. We don't have enough to subtract away another group of 2, so we say that 1 is the leftover amount or remainder from the division.

11 ÷ 2 = 5 Remainder 1

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