Division of Integers: Arithmetic operation is the branch of mathematics that involves the addition, subtraction, division, and multiplication of all types of real numbers, including integers. Integers are specific numbers that include negative numbers, positive numbers, and zero, but no fractions. The division is the inverse process of multiplication. 

In this article, we will define the division of integers, rules of division, formulas, and their applications.

What is an Integer?

The term “integer” was taken in Mathematics from the Latin word “integer”, which means intact or whole. Integers include whole numbers and negative whole numbers, i.e. integers can be positive, negative, or zero.

Definition: An integer is a number that doesn’t have the decimal or the fractional part from the set of negative and positive numbers, including zero.

Examples: of integers are: $-5,0,1,5,8,97,$ and $3043.$

Please note that a set of integers, defined as $Z,$ includes:
1. Positive Integers: An integer is a positive integer if it is greater than zero. 
Example: $1,2,3...$
2. Negative Integers: An integer is a negative integer if it is less than zero.
Example: $-1,-2,-3...$
3. Zero is defined as neither negative nor positive integer.
$Z=\{\dots –7,–6,–5,–4,–3,–2,–1,0,1,2,3,\dots \}$

Definition of Division of Integers

Definition: The division of integers is the opposite operation of the multiplying of integers. It is the process by which one tries to determine how many times a number is contained into another.

Dividing $20$ by $5$ means finding an integer that, when multiplied with $5$ gives us $20.$ Such an integer is $4.$
Therefore, we write $20÷5=4$ or, $\frac{20}{5}=4$. Similarly, dividing $36$ by $-9$ means finding an integer which, when multiplied with $-9$ gives $36.$ Such an integer is $-4.$

Therefore, we write $36÷(–9)=–4$ or, $\frac{36}{–9}=–4$

Dividing $(-35)$ by $(-7)$ means getting an integer that, when multiplied with $(-7)$ gives $(-35).$
Such an integer is $5.$ 
Therefore, $\left(35\right)÷(–7)=5$ or, $\frac{–35}{–7}=5$

Dividend: The number to be divided is known as a dividend.

Divisor: The number which divides is known as the divisor.

Quotient: The result of division is known as the quotient.

Remainder: If a number is is not completely divisible by the divisor, the left out part of the dividend, which is less than the divisor, is called the remainder.

Example: If we divide $26$ by the number $6,$ the dividend is $26,$ the divisor is $6,$ the quotient is $4,$ and the remainder is $2.$

Rules of Division of Integers

It follows from the above discussion that when a dividend is negative, and the divisor is negative, the quotient is positive. When the dividend is negative, and the divisor is positive, the quotient is negative.

How to Solve the Division of Integers?

Use the following rules for the division of integers:

Rule 1: The quotient of the two integers, either both positive or both negative, is a positive integer equal to the quotient of the corresponding fundamental values of the integers.
Thus, for dividing two integers with like signs, we divide their values regardless of their sign and give plus sign to the quotient.

Rule 2: The quotient of a positive and a negative integer is a negative integer. The absolute value is equal to the quotient of the corresponding absolute values of the integers.
Thus, we divide their values regardless of their sign and give minus sign to the quotient for dividing integers with unlike signs.

Division of Integers Formula

The formulas of the division of integers are given below in the table:

Type of  Numbers	Operation	Result	Example
Positive Positive	Divide	Positive $(+)$	$12÷6=2$
Negative Negative	Divide	Positive $(+)$	$-12-6=2$
Positive Negative	Divide	Negative $(-)$	$12÷(-6)=-2$
Negative Positive	Divide	Negative $(-)$	$-12÷6=-2$

Same as the multiplication, you have to divide the integers without the sign, then give the symbol according to the rule as given in the table. The division of two integers with the like signs gives a positive quotient, and the division of two integers with unlike signs gives a negative quotient.

Properties of Division of Integers 

There are some of the properties of a division of integers which are given below:

1. If $a$ and $b$ are integers, then a÷b is not necessarily an integer. For example, $14÷2=7.$ Here, the quotient is an integer.
But, in $15÷4,$ we observe that the quotient is not an integer. Here, the result is $\frac{15}{4}=3_4^3$. the quotient is $3,$ and the remainder is $3$

2. If $a$ is an integer other than $0,$ then a÷a=1.

3. For every integer $a,$ we have $a÷1=a.$

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

5. If $a$ is an integer, then $a÷0$ is not meaningful.

6. If $a,b,c$ are integers, then
$a>b⟹a÷c>b÷c,$ if $c$ is positive.
$a>b⟹a÷c<b÷c,$ if $c$ is negative.

Let us understand the division of integers method with the help of some solved examples.

Solved Examples – Division of Integers

Q.1. In a test, $(+5)$ marks are awarded for every correct answer and $(-2)$ are provided for every incorrect answer. Radhika answered all the questions and scored $30$ marks though she got $10$ correct answers. Find the number of incorrect answers.
Ans: Marks awarded for each correct answer $=5$
So, marks allotted for $10$ correct answers $=5\times 10=50$
Radhika’s score $=30$
Marks obtained for incorrect answers$=30-50=-20$
Penalty for each wrong answer $=(-2)$
Hence, number of incorrect answers $=(-20)÷(2)=10$

Q.2. In a test, $(+5)$ marks are given for every correct answer and $(-2)$ are provided for every incorrect answer. Jay answered all the questions and scored $(-12)$ marks though he got $4$ correct answers. How many wrong answers had they attempted?
Ans: Marks awarded for each correct answer $=5$
So, marks allotted for $4$ correct answers $=5\times 4=20$
Jay scored $=(-12)$
Marks obtained for incorrect answers $=(-12)-20=-32$
Marks are given for one wrong answer $=(-2)$
Therefore, number of incorrect answers $=(-32)÷(-2)=16$

Q.3. A shopkeeper earns a profit of $\text{Math input error}$ by selling one pen and incurs a loss of $40$ paise per pencil while selling pencils of her old stock. In a particular month, she incurs a loss of $\text{Math input error}$ In this period, she sold $45$ pens. How many pencils did she sell in this period?
Ans: Profit earned by selling one pen $\text{Math input error}$
Profit earned by selling $45$ pens $\text{Math input error}$ which we denote by $\text{Math input error}$
Total loss $\text{Math input error}$ which we denote by (-₹5)
Profit earned $+$ Loss incurred $=$ Total loss
Therefore, loss incurred = Total Loss – Profit earned.
$\text{Math input error}$ paise.
Loss incurred by selling one pencil $=40$ paise, which we write as $-40$ paise
So, the number of pencils sold $=(-5000)÷(40)=125.$

Q.4. A shopkeeper earns a profit of $\text{Math input error}$ by selling one pen and incurs a loss of $40$ paise per pencil while selling pencils of her old stock. In the next month, she earns neither profit nor loss. If she sold $70$ pens, how many pencils did she sell?
Ans: In the next month there is neither profit nor loss.
So, Profit earned $+$ Loss incurred $=0$
o.e., Profit earned $=-$ Loss incurred.
Now, profit earned by selling $70$ pens $\text{Math input error}$
Hence, loss incurred by selling pencils $\text{Math input error}$ which we indicate by $\text{Math input error}$ or $-7,000$ paise.
Total number of pencils sold $=(-7000)÷(-40)=175$ pencils.

Q.5. Find the value of: $[32+2\times 17+-6]÷15$
Ans: We have,
$[32+2\times 17+-6]÷15$
$=[32+34+(–6)]÷15=(66–6)÷15=60÷15=\frac{60}{15}=4$
The answer is $4.$

Q.6. Find the quotient and the remainder of $57÷6.$
Ans: When we divide $57$ by $6,$ the quotient is $9,$ and the remainder is $3.$

Summary

In this article, we discussed the definition of integers and understood the division of integers with some examples. The division of integers formulas and the properties learned will help in solving the questions quickly. The solved sample questions will help understand how to solve the division of integers.

Frequently Asked Questions (FAQ) – Division of Integers

Q.1. How do you divide integers step by step?
Ans: To divide the integers step by step, follow the given method:
Divide their fundamental values
Then, determine the sign of the final answer (quotient) by using the given conditions.
If the symbol of the two integers is the same, the quotient will be a positive integer.
$(+)÷(+)=+$
$(-)÷(-)=+$
If the sign is different for both the integers, then the quotient will be a negative integer.
$(+)÷(-)=-$
$(-)÷(+)=-$

Q.2. What is the rule of division of integers?
Ans: The rules of the division of integers are given below:
1. The quotient of two integers, either both positive or both negative, is a positive integer equal to the quotient of the corresponding absolute values of the integers.
2. The quotient of a positive and a negative is a negative integer, and its absolute value is equal to the quotient of the corresponding fundamental values of the integers.

Q.3. How do you divide integers in $7^{\mathrm{t}\mathrm{h}}$ grade?
Ans: The rules that are provided in the above article are used to divide the integers in the $7^{\mathrm{t}\mathrm{h}}$ grade.

Q.4. What are the rules of integers?
Ans: The rules we have for the integers are:
1. The sum of the two positive integers is an integer.
2. The sum of the two negative integers is an integer.
3. The product of two positive integers is an integer.
4. The product of two negative integers is an integer.
5. The sum of an integer and its additive inverse is equal to the number zero.
6. The product of an integer and its reciprocal is equal to the number $1.$

Q.5. How to divide integers? Give an example.
Ans: The example of dividing integers is given below:
Example: Divide $-98$ by $-14$
$-98÷(-14)$
$\frac{–98}{–14}=\frac{98}{14}=7$
Hence, the required answer is $7.$

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