In arithmetic addition, subtraction, multiplication, and division are four basic operations used in solving problems. If you know and follow the basic rules of integers in multiplication and division, the process of solving the problems becomes smooth and easy.

## Multiplication and Division of Integers

Multiplication and division are nothing but respectively the repeated addition and subtraction of numbers. You can perform multiplication and division by using repeated addition or subtraction of integers and applying their rules in integers.

But the process of repeated addition or subtraction is long and time-consuming. As in the case of the addition or subtraction of integers, for multiplying and dividing integers also there are certain rules.

### Multiplication of Integers

Multiplication can be considered as repeated addition (accumulation). There are certain rules for the multiplication of integers.

The rules of multiplication of integers are:

- Multiplication between two positive numbers
- Multiplication between two negative numbers
- Multiplication between a positive number and a negative number

Now, coming to the result of the multiplication of two integers, any of the following can happen:

- The product of two integers with similar sign numbers will always be positive. This means
- the product of two positive integers is positive
- the product of two negative numbers is positive

- The product of two integers with different sign numbers will always be negative. The product of a positive number and a negative number (integers with different signs) will always be negative.

**Note:** The product of two integers is always an integer.

The above rules of integers in multiplication can be summarized as

Type of Numbers | Result | Example |

Positive $\times$ Positive | Positive $\left(+ \right)$ | $3 \times 7 = 21$ |

Negative $\times$ Negative | Positive $\left(+\right)$ | $\left(-3 \right) \times \left(-7 \right) = 21$ |

Positive $\times$ Negative | Negative $\left(- \right)$ | $3 \times \left(-7 \right) = -21$ |

Negative $\times$ Positive | Negative $\left(- \right)$ | $\left(-3 \right) \times 7 = -21$ |

**Note:** In the case of multiplication of integers, just multiply the numbers without the sign. Once the product is obtained give the sign according to the rule of multiplication given in the above table.

### Steps in Multiplication of Integers

The following steps are performed while multiplying two integers:

**Step 1:** Determine the absolute value of the numbers.

**Step 2:** Find the product of the absolute values.

**Step 3:** Once the product is obtained, determine the sign of the number according to the rules or conditions.

**Note:** Absolute value of an integer, is an integer without any sign.

For example,

- the absolute value of $-12$ is $12$
- the absolute value of $+12$ is $12$

### Division of Integers

As multiplication is repeated addition (accumulation), similarly division is repeated subtraction(distribution) of numbers. Hence, dividing integers can be considered as an operation opposite to that of multiplication. Still, the rules for the division of integers are the same as multiplication rules.

**Note:** The result of two integers is not always an integer.

- When the remainder is 0, then the result is an integer.
- When the remainder is not 0, then the result is not an integer.

The rules of division are

- The result of two positive integers will always be positive.
- The result of two negative integers will always be positive.
- The result of a positive integer and a negative integer will always be negative.

Type of Numbers | Result | Example |

Positive $\div$ Positive | Positive $\left(+ \right)$ | $15 \div 3 = 5$ |

Negative $\div$ Negative | Positive $\left(+ \right)$ | $\left(-15 \right) \div \left(-3 \right) = 5$ |

Positive $\div$ Negative | Negative $\left(- \right)$ | $15 \div \left(-3 \right) = -5$ |

Negative $\div$ Positive | Negative $\left(- \right)$ | $\left(-15 \right) \div 3 = -5$ |

As in the case of multiplication, divide the integers without the sign, then give the sign according to the rule as given in the above table.

### Steps in Division of Integers

The following steps are performed while multiplying two integers:

**Step 1:** Determine the absolute value of the numbers.

**Step 2:** Find the product of the absolute values.

**Step 3:** Once the product is obtained, determine the sign of the number according to the rules or conditions.

**Note:** Absolute value of an integer, is an integer without any sign.

For example,

- the absolute value of $-12$ is $12$
- the absolute value of $+12$ is $12$

## Multiplication and Division of Integers Using Number Line

As discussed earlier, multiplication and division are the repeated addition and subtraction of numbers. You can perform multiplication and division by using rules of addition and subtraction using the number line.

### Multiplication of Integers Using Number Line

To perform multiplication on a number line, we start from zero $\left(0 \right)$ and move towards the right side of the number line for a given number of times.

Let us consider an example. Multiply $2 \times 4$ using a number line.

Starting from zero, four groups of equal intervals of $2$ are formed on the number line. This way we will reach 8.

The following number line shows $2 \times 4 = 8$.

#### Multiplication of Positive Numbers on Number Line

When multiplying two or more positive numbers, there is a simple rule that follows simple multiplication. Since this is multiplication, we will move towards the right side of the number line. Let’s solve $3 \times 5$.

Starting from zero, five groups of equal intervals of $3$ are formed on the number line. This way we will reach $15$.

The following number line shows $3 \times 5 = 15$.

#### Multiplication of Negative Numbers on Number Line

When multiplying more than two negative numbers, use the Even-Odd Rule:

- Count the number of negative signs.
- if the number of negatives is even, the result is positive
- if the number of negatives is odd, the result is negative

**Note: **

- Numbers that come in a multiplication table of $2$ are called even numbers such as $2, 4, 6, 8, 10, 12, …$
- Numbers that do not come in a multiplication table of 2 are called odd numbers such as $1, 3, 5, 7, 9, 11, …$
- Every natural number is either an even number or an odd number.

Let’s consider an example of $-4 \times 4$.

The number of negative signs is $1$, which is odd. So, the result will be negative (and will lie on the left side of $0$(zero)).

Starting from zero, four groups of equal intervals of $4$ are formed on the left side of the number line. This way we will reach $-16$.

### Division on Number Line

As mentioned above division is nothing but a repeated subtraction. We will use this fact and understand the representation of division operation on a number line.

Consider the number $x$ being divided by $y$.

Since division operation can be performed as repeated subtraction, therefore we will perform the subtraction operation by moving towards the left on the number line.

Let’s understand the steps for division on a number line.

**Step 1:** Draw a number line, plotting the multiples of $y$ starting from $0$ and mark the dividend $x$ on the number line. We will take $x$ as the reference.

**Step 2:** Starting from $x$, we keep subtracting groups of $y$ units each until we reach the number $0$. Each time we do the subtraction, we move by y units towards the left of $x$ until we reach $0$. The alternate method to do this is starting from $0$ as the reference point, we can move towards the right by groups of $y$ units each until we reach the number $x$.

**Step 3:** The number of steps of $y$ units each that we moved towards the left of $x$ to reach $0$ or the number of steps of $y$ units each that we moved towards the right of $0$ to reach $x$ will give us the quotient.

Let’s take the example of dividing $12$ by $6$ to understand the representation of division on a number line.

Let us follow the steps that we discussed above to perform the division $12 \div 6$.

To perform $12 \div 6$, we will make a number line and plot the first few multiples of $6$ starting from $0$ which includes the dividend $12$.

Encircle the dividend $12$.

Starting from $12$, move towards the left by $6$ units and keep repeating these moves of $6$ units each until we reach $0$.

The number of moves made to reach $0$ from $14$ will be the quotient. As we see that, we had to move by $2$ jumps to reach $0$.

Therefore, the quotient is $2$.

#### Division of Negative Numbers on a Number Line

We will now look into the representation of the division of negative numbers on a number line. The steps to show the division will be very similar as discussed in the above section. Let’s take two cases with examples.

**Case I: Negative number divided by a positive number**

Example: $\left(-12 \right) \div 3$

As per the rules of integers operation when a negative number is divided by a positive number or vice-versa a result is always a negative number.

We plot the negative multiples of $3$ on the number line starting from $0$ to at least $-12$ as the dividend is $-12$.

Encircle the dividend $-12$.

Starting from $-12$, move towards the right by $3$ units and keep repeating these moves of $3$ units each until we reach $0$.

We see that there are a total of $4$ jumps made each consisting of $3$ units to reach $-12$.

Since the jumps are made towards the left of $0$ on the number line, hence the result is $-4$.

Therefore, $\left(-12 \right) \div 3 = -4$.

**Case II: Negative number divided by a negative number**

Example: $\left(-15 \right) \div \left(-3 \right)$

As per the rules of integer operation when a negative number is divided by a negative number, the result is always a positive number.

We plot the positive multiples of $3$ at least to $15$ as shown in the diagram.

Encircle the dividend $15$.

Starting from $15$, move towards the left by $3$ units and keep repeating these moves of $3$ units each until we reach $0$.

We see that there are a total of $5$ jumps made each consisting of $3$ units to reach 0.

Therefore, $\left(-15 \right) \div \left(-3 \right) = 5$.

#### Division on a Number Line with Remainders

While dividing numbers, we come across a lot of situations wherein the dividend is not completely divisible by the divisor. This happens when the dividend is not a multiple of the divisor or the divisor is not a factor of the dividend.

When this situation arises, we get a non-zero remainder.

We will now understand how to represent the division on a number line with remainders by taking an example.

Let’s divide the number $12$ by $5$ and represent it on a number line.

Since the divisor is $5$, we start making groups of $5$ units and start moving towards the right of $0$ starting from $0$ to reach the number $12$.

Encircle the dividend $17$.

Start making groups of $5$ from $0$.

We see that the first group is $0$ to $5$, the second group is $5$ to $10$, and the third group is $10$ to $15$. Thus, we have three groups of $5$ from $0$ to $15$.

Note that, for the third group we need a minimum of $5$ more units but we have only $2$ more units that are from $15$ to $17$.

Hence, we can say that the remainder is $2$ as these $2$ units are not forming a group of $5$.

Therefore, the quotient is $3$ since there are three groups of $5$ and the remainder is $2$.

Hence, we can represent this division as $17 \div 5 =$ Quotient is $2$ and Remainder is $5$.

## Key Takeaways

- If the signs are different the answer is negative.
- If the signs are alike the answer is positive.

## Practice Problems

- $\left( +8 \right) \times \left(-1 \right)$
- $\left(+4 \right) \times \left(+6 \right)$
- $\left(-3 \right) \times \left(-7 \right)$
- $\left(-9 \right) \times \left(-2 \right)$
- $\left( -11 \right) \times \left(+5 \right)$
- $\left(+12 \right) \times \left(0 \right)$
- $\left(-20 \right) \times \left(-4 \right)$
- $\left(-15 \right) \times \left(+13 \right)$
- $\left(-20 \right) \times \left(-5 \right)$
- $\left(-9 \right) \times \left(+4 \right)$
- $\left(+3 \right) \times \left(-6 \right)$
- $\left(+11 \right) \times \left(0 \right)$
- $\left(-7 \right) \times \left(-4 \right)$
- $\left(-9 \right) \times \left(+8 \right)$
- $0 \div \left(-8 \right)$
- $\left(+9 \right) \div \left(-3 \right)$
- $\left(-11 \right) \div \left(-6 \right)$
- $\left(-12 \right) \div \left(-3 \right)$
- $\left(-25 \right) \div \left(+5 \right)$
- $\left(+45 \right) \div \left(-3 \right)$
- $\left(-24 \right) \div \left(-4 \right)$
- $\left(-78 \right) \div \left(13 \right)$
- $\left(30 \right) \div \left(-10 \right)$
- $\left(+14 \right) \div \left(-2 \right)$
- $\left(-16 \right) \div \left(-8 \right)$
- $\left(-48 \right) \div \left(+6 \right)$
- $\left(4 \right) \div \left(2 \right)$
- $0 \div \left(6 \right)$