# Multiplication Of Integers

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We can add and subtract integers. Let us now learn how to multiply integers.

## Multiplication of a Positive and a Negative Integer

We know that multiplication of whole numbers is repeated addition. For example,

5 + 5 + 5 = 3 × 5 = 15

Can you represent addition of integers in the same way?

We have from the following number line, **(–5) + (–5) + (–5) = –15**

Now, **try this out yourself.**

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### Try These

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But we can also write:

(–5) + (–5) + (–5) = 3 × (–5)

Therefore,

3 × (–5) =

Similarly

(– 4) + (– 4) + (– 4) + (– 4) + (– 4) = 5 × (– 4) =

Now, try it again by yourself.

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We see that: When a number is added to itself, a certain number of times - we get the resulting number to be equal to:

**Resulting Number** = **Number of repetitions** × **Number Value**

So:

Again,

(–7) + (–7) + (–7) =

Now, let us see how to *find the product of a positive integer and a negative integer without using number line.*

Let us find 3 × (–5) in a different way. First, find 3 × 5 and then put minus sign (–) before the product obtained. We get –

Similarly, 5 × (– 4) = –(5 × 4) =

**Find the products of:**

4 × (– 8) = -(

3 × (– 7) = -(

6 × (– 5) = -(

2 × (– 9) = -(

10 × (– 43) = -(

Till now we multiplied integers as (positive integer) × (negative integer). Now, let's try multiplying them as (negative integer) × (positive integer).

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## Try These

**1. Find:**

(i) 6 × (–19) =

(ii) 12 × (–32) =

(iii) 7 × (–22) =

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We first find: –3 × 5.

Observe the pattern as we move along.

- Now, 2 × 5 =
- 1 x 5 =
- 0 x 5 =
- We notice that as we decrease the given integer, the multiple of the integer with the constant i.e. 5 can also be written as the subtraction of 5 from the preceeding multiple.
- Likewise, -1 x 5 =
- -2 x 5 =
- -3 x 5 =
- This pattern is useful in understanding how addition, subtraction and multiplication are co-related.

So, we get

Using such patterns, we also find that:

We thus find that while multiplying a positive integer and a negative integer,

**multiply them as whole numbers and put a minus sign (–) before the product. We thus, get a negative integer.**

## Try These

**1. Find:**

(a) 15 × (–16) =

(b) 21 × (–32) =

(c) (– 42) × 12 =

(d) –55 × 15 =

**2. Check if**

(a) 25 × (–21) = (–25) × 21 =

(b) (–23) × 20 = 23 × (–20)=

**Write five more such examples.**

1.17×(−15) = (

2.35×(−42) = (

3.(−56)×17 =

4.64×(−29)=(

5.49×(−81)=(

In general, for any two positive integers *a* and *b* we can say:

**a** × **(– b)** = **(– a)** × **b** = –(**a** × **b**)

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## Multiplication of two Negative Integer

Similarly, let's try to understand the pattern when dealing with the multiplication of two negative integers.

What is the product of (–3) × (–2) ? Let's find out.

Observe the below pattern:

- Now, -3 × 4 =
- -3 x 3 =
- -3 x 2 =
- -3 x 1 =
- -3 x 0 =
- -3 x (-1) =
- -3 x (-2) =
- This pattern is useful in understanding how addition, subtraction and multiplication are co-related.

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Do you see any pattern? Observe how the products change. Based on this observation, complete the following:

Finding the product of

- Now, -4 × 4 =
- -4 x 3 =
- -4 x 2 =
- -4 x 1 =
- -4 x 0 =
- -4 x (-1) =
- -4 x (-2) =
- -4 x (-3) =
- And the pattern continues

## Try These

(i) Starting from (–5) × 4, find (–5) × (– 6) =

(ii) Starting from (– 6) × 3, find (– 6) × (–7) =

From these patterns we can find out that,

(–3) × (–1) =

(–3) × (–2) =

(–3) × (–3) =

Similarly,

(– 4) × (–1) =

So,

(– 4) × (–2) = 4 × 2 =

(– 4) × (–3) = 4 × 3 =

So, observing these products we can say that:

**The product of two negative integers is a positive integer. We multiply the two negative integers as whole numbers and put the positive sign before the product.**

Thus, we have

(–10) × (–12) =

(–15) × (– 6) =

In general, for any two positive integers a and b,

**(-a)** × **(– b)** = **a** × **b**

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## Try These

**1. Find:**

a.(–31) × (–100) =

b.(–25) × (–72) =

c.(–83) × (–28) =