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While studying mathematics we notice some arithmetic operation that includes the processes of addition, subtraction, division, and multiplication.

Integers are the whole numbers that are non-fractional.

In integers, we can find,

The numbers which we count (1,2,3….).

The number 0

The negative numbers ( -1, -2, -3,...)

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Multiplication is the addition of numbers but the rules for the multiplication of integers are different from the addition of integers.

The only thing that makes the multiplication and division different is the sign. You need to be very careful regarding the signs while doing multiplication and division.

In the multiplication of integers, you have the following properties:

### Closure Property:

The multiplication value of two integers is always an integer.

According to closure property if you multiply two integers suppose p x q then the product of p x q is also an integer.

Here, p x q will be an integer, for each integer p and q.

### Commutative Property:

In case of any two integers m and n,

m×n = n×m

### Associative Property:

multiplication between integers is associative. i.e., for any three integers a, b, c, we have: a × ( b × c) = (a × b) × c

### Distributive Property:

The multiplication of integers is distributive over their addition. For example, if we have three integers, 1, 2, 3.

1 x (2 + 3) = (1 x 2) + (1 x 3)

Therefore, multiplication is distributive over the addition of integers.

### Multiplication by Zero:

When any integer is multiplied by Zero we get zero as the product.

For example: if we have an integer 5

5×0 = 0

### Multiplication by Identity:

When you multiply any integer by 1 you get 1 itself as the result.

1’ is the multiplicative identity for integers.

In general, for an integer 3, we have

3 x 1 = 1 x 3 = 3

Rule 1: When Multiplication occurs between two positive integers the product is always positive.

For example: 2×2 = 4

Rule 2: When Multiplication occurs between two negative integers the product is always positive.

For example: (-2)×(-2) = 4

Rule 3: The product of a positive integer and a negative integer is negative.

For example: 2×(-2) = - 4

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Question: Find the product of the following:

(i) (–18) × (–10) × 9

(iii) (–5) × (–2) × (– 8) × (– 7)

Solved:

(i) (–18) × (–10) × 9 = [(–18) × (–10)] × 9 = 180 × 9 = 1620

(iii) (–5) × (–2) × (– 8) × (– 7) = [(–5) × (–2)] × [(– 8) × (– 7)] = 10 × 56 = 560

When you distribute integers you are carrying out the function division of integers. The process of dividing integers is exactly the opposite of multiplying integers.

In both, cases multiplying the integer or dividing the integer the rules are quite similar. But it’s not necessary to always find integers as your quotient value.

In the division of integers, you have the following properties:

Suppose a and b are two integers, then a ÷ b does not necessarily have to be an integer.

If an integer a is not equals to 0, then a ÷ a = 1.

For every integer a, you have a ÷ 1= x.

If an integer a is non-zero, then 0 ÷ a = 0.

If a is an integer, then a ÷ 0 is not valid.

Suppose we have three integers x, y, which are non-zero integers, then (x ÷ y) ÷ z ≠ x÷ (y ÷ z), unless z = 1.

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

One of the things to always remember is when you are dividing you should always divide without the signs but after getting the solution of the integer gives the sign according to the sign given in the problem.

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Question: solve- 91 ÷ 7, -117 ÷ 13, -98 ÷ -14

Solution:

91 ÷ 7

= 91/7

=13

-117 ÷ 13

= -117/13

= -9

-98 ÷ (-14)

= -98/-14

= 98/14

= 7

FAQ (Frequently Asked Questions)

1. Why is it Important to Use Proper Sign Convention for Solving the Multiplication and Division of Integers?

Ans: It is important to use to find the proper solutions for the multiplication and division of integers. While using the proper sign we will get the correct answer to every problem we solve.

A single wrong sign can deduct your marks by proving that solution as wrong.

Suppose a problem is given is to you which has two integers in the problem one positive and one negative the answer should be in negative

Let us see take an example for a better understanding

By solving 2 x (-4) = -8

But we write the answer as 8 it will be incorrect because you have not provided the correct sign convention for this problem.

It is always good to keep in your mind to use proper sign convention.

2. Why is it Required to Follow the Rules of Division and Multiplication of Integers?

Ans: Rules are introduced to keep the solving method easier which will reduce the amount of confusion we create while solving sums.

Any alternative way or any properties or rules is beneficial for our good to solve sums.

The rules of multiplication and division of integers include some ways or formulas which are key to simplify the problems in much easier ways but there is always an exception.

So, no problem is specifying one rule. Every rule has its different ways to solve a different rule to solve. Rules are provided to use to find correct answers with a hustle free method.

By using rules in multiplication and division of integers we get to know that if we multiply positive to positive we find the solution in positive.