### Key Concepts

- Divide integers with different signs
- Divide integers with the same sign
- Write equivalent quotients of integers

**1.6 Division of Integers**

**Introduction:**

Division is the inverse operation of multiplication.

Let us see an example for whole numbers.

Dividing 20 by 5 means finding an integer which when multiplied with 5 gives us 20, such an integer is 4.

Since 5 × 4 = 20

So, 20 ÷ 5 = 4 and 20 ÷ 4 = 5

Therefore, for each multiplication statement of whole numbers, there are two division statements.

**Properties of division of integers: **

**Closure under division**

Division of integers does not follow the closure property.

Let’s consider the following pairs of integers.

(−12) ÷ (−6) = 2 (Result is an integer)

(−5) ÷ (−10) =

1/2 (Result is not an integer)

We observe that integers are not closed under division.

**Commutative property of division**

Division of integers is not commutative for integer.

Let’s consider the following pairs of integers.

(–14) ÷ (–7) = 2

(–7) ÷ (–14) = 1/2

(–14) ÷ (–7) ≠ (–7) ÷ (–14)

We observe that division is not commutative for integers.

**Division of an integer by zero**

Any integer divided by zero is meaningless.

Example: 5 ÷ 0 = not defined

Zero divided by an integer other than zero is equal to zero.

Example: 0 ÷ 6 = 0

**Division of an integer by 1**

When we divide an integer by 1 it gives the same integer.

Example: (– 7) ÷ 1 = (– 7)

This shows that negative integer divided by 1 gives the same negative integer. So, any integer divided by 1 gives the same integer.

In general, for any integer *a*, *a* ÷ 1 = *a*

**Rules for the division of integers:**

**Rule 1: **The quotient value of two positive integers will always be a positive integer.

**Rule 2: ** For two negative integers the quotient value will always be a positive integer.

**Rule 3:** The quotient value of one positive integer and one negative integer will always be a negative integer.

The following table will help you remember rules for dividing integers:

Types of Integers | Result | Example |

Both Integers Positive | Positive | 16 ÷ 8 = 2 |

Both Integers Negative | Positive | –16 ÷ –8 = 2 |

1 Positive and 1 Negative | Negative | –16 ÷ 8 = –2 |

**1.6.1 Division of Integers with Different Signs**

**Steps:**

- First divide them as whole numbers.

- Then put a minus sign (–) before the quotient. We, thus, get a negative integer.

**Example:**

(–10) ÷ 2 = (– 5)

(–32) ÷ (8) = (– 4)

In general, for any two positive integers *a* and *b*, *a* ÷ (– *b*) = (– *a*) ÷ *b* where* b* ≠ 0.

**1.6.2 Division of Integers with the Same Sign**

**Steps:**

- Divide them as whole numbers.

- Then put a positive sign (+). That is, we get a positive integer.

**Example: **

(–15) ÷ (–3) = 5

(–21) ÷ (–7) = 3

In general, for any two positive integers *a* and *b*, (– *a*) ÷ (– *b*) = *a* ÷ *b* where *b* ≠ 0.

**1.6.3 Write Equivalent Quotients of Integers **

What is Quotient in division?

The number obtained by dividing one number by another number.

If *p* and *q *are integers, then

−(p / q)

=-p/q =p/−q.

**Example:** Show that the quotients of

−(18 / 4) , −18 / 4, and 18/−4 are equivalent.

−(18/4) = −(18÷4)=−(4.5)= −4.5

−18 / 4 = −18÷4 = −4.5

18 / −4 = 18÷−4 = −4.5

∴The quotients of

−(18 / 4) , −18 / 4, and 18/−4 are equivalent to −4.5

**Example: **Evaluate [(– 8) + 4)] ÷ [(–5) + 1]

**Solution:** [(– 8) + 4)] ÷ [(–5) + 1]

= (−4) ÷ (−4)

= 1

**Example:** Verify that *a* ÷ (*b* + *c*) ≠ (*a* ÷ *b*) + (*a* ÷ *c*) when *a* = 8, *b *= – 2, *c* = 4.

**Solution:** L.H.S = *a* ÷ (*b* + *c*)

= 8 ÷ (−2 + 4)

= 8 ÷ 2 = 4

R.H.S = (*a* ÷* b*) + (*a* ÷ *c*)

= [8 ÷ (−2)] + (8 ÷ 4)

= (−4) + 2

= −2

Here, L.H.S ≠ R.H.S

Hence verified.

## Exercise

- $4000 is distributed among 25 women for the work completed by them at a construction site. Calculate the amount given to each woman.
- 66 people are invited to a birthday party. The suppliers have to arrange tables for the invitees. 7 people can sit around a table. How many tables should the suppliers arrange for the invitees?
- Calculate the number of hours in 2100 minutes.
- When the teacher of class 6 asked a question, some students raised their hands. But instead of raising one hand, each student raised both their hands. If there are 56 hands in total, how many students raised their hands?
- Emma needs 3 apples to make a big glass of apple juice. If she has 51 apples, how many glasses of juice can she make?
- Simplify –40 ÷ (–5)
- Find the quotient of .
- Why is the quotient of two negative integers positive?
- Classify the quotient –50 ÷ 5 as positive, negative, zero or undefined.
- Simplify –28 ÷ (–7).

### What we have learned

- Restate that the quotient of two integers with unlike signs is a negative integer.
- Restate that the quotient of two integers with like signs is a positive integer.
- Perform division of two integers with like signs.
- Perform division of two integers with unlike signs.
- Describe the procedure for dividing integers with like signs.
- Describe the procedure for dividing integers with unlike signs.
- Apply the procedures for integer division to complete the exercises.