It is important for students to learn the Properties of Division of Integers as we are familiar with natural and whole numbers. We know that when a smaller whole number is subtracted from a larger one, we get a whole number. But what about \(2 – 6,5 – 7,12 – 17,\) etc.? There are no whole numbers to represent them. So there is a need to extend our whole number system to contain numbers to represent the above differences.

Similar to (1,2,3,4,5,6, \ldots ) we introduce new numbers denoted by ( – 1, – 2, – 3, – 4, – 5, – 6, \ldots ) called minus one, minus two, minus three, minus four, minus five and so on respectively such that (1 + \left({ – 1} \right) = 0,2 + \left({ – 2} \right) = 0,3 + \left({ – 3} \right) = 0,) and so on.

Integers

All natural numbers, \(0\) and negatives of counting numbers, are called integers. Thus, \( \ldots \ldots . – 4, – 3, – 2, – 1,0,1,2,3,4, \ldots \ldots .,\) etc., are all integers.

Positive Integers: \(1,2,3,4,5, \ldots \ldots \) etc., are all positive integers.

Negative Integers: \( – 1, – 2, – 3, – 4, – 5, \ldots \ldots \) are all negative integers.

Zero: It is an integer that is neither positive nor negative.

Similar to operations on whole numbers, we have the below operations in integers.

1. Addition of Integers
2. Subtraction of Integers
3. Multiplication of Integers
4. Division of Integers

Addition of Integers

If two positive integers are added, the addition is done exactly the same way we add natural numbers or whole numbers. To add a positive and a negative integer, we find the difference between their numerical values regardless of their signs and give the sign of the integer with the greater value to it.
Example: \(36 + 27 = 63\) and \( – 47 + 18 = – 29\)

Subtraction of Integers

Subtraction is the inverse process of addition. To subtract one integer from another, we take the additive inverse of the integer to be subtracted and add it to the other integer. Thus, if a and b are two integers then \(a – b = a + \left({ – b} \right).\)
Example: \(2 – 7 = – 5\) and \( – 5 – \left({ – 7} \right) = – 5 + 7 = 2\)

Multiplication of Integers

To find the product of two integers with unlike signs, we find the product of their values regardless of their signs and give a minus sign to the product. To find the product of two integers with the same sign, we find the product of the values of their signs and give a plus sign to the product.
Example: \( – 2 \times – 3 = 6\) and \(6 \times \left({ – 4} \right) = – 24\)

Now, let us see the division of integers in detail.

Division of Integers

We know that division of whole numbers is an inverse process of multiplication. In this article, we shall extend the same idea to integers.
We know that dividing \(8\) by \(4\) means finding an integer that multiplied with \(4\) gives us \(8.\) Such integer is \(2.\) Therefore, we write \(8 \div 4 = 2\) or \(\frac{8}{4} = 2.\)
Similarly, dividing \(24\) by \(-6\) means finding an integer which, when multiplied with \(-6,\) gives \(24.\)Such an integer is \(-4.\) So, we write \(24 \div \left({ – 6} \right) = – 4\) or \(\frac{{24}}{{ – 6}} = – 4.\)
Dividing \(\left({ – 15} \right)\) by \(\left({ -3} \right)\) means what integer should be multiplied with \( – 3\) to get \( – 15.\) Such an integer is \(5.\) Therefore, \(\left({ – 15} \right) \div \left({ – 3} \right) = 5\) or \(\frac{{ – 15}}{{ – 3}} = 5.\)
We have below definitions regarding the division:
Dividend: The number to be divided is called the dividend.
Divisor: The number which divides is called the divisor.
Quotient: The result of division is called the quotient.

Rules for Division of Integers

When the 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.

Thus, we have the below rules for the division of integers.

Rule 1:The quotient of two integers, both positive or negative, is a positive integer equal to the quotient of the corresponding absolute values of the integer.

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. Its absolute value is equal to the quotient of the corresponding absolute values of the integer.

Thus, we divide their values regardless of their sign and give minus sign to the quotient for dividing integers with unlike signs.

What are Properties of Division of Integers?

Division of integers has the below properties:

1. If \(a\) and \(b\) are integers, then \(a \div b\) is not necessarily an integer.
Example: \(15 \div 2, – 12 \div 5\) are not integers.
2. If \(a\) is an integer different from \(0,\) then \(a \div a = 1.\)
Example: \(5 \div 5 = 1\) and \( – 7 \div – 7 = 1\)
3. For every integer \(a,\) we have \(a \div 1 = a.\)
Example: \(8 \div 1 = 8\) and \( – 15 \div 1 = – 15\)
4. If \(a\) is a non-zero integer, then \(0 \div a = 0\)
Example: \(0 \div 4 = 0\) and \(0 \div 3 = 0\)
5. If \(a\) is an integer, then \(a \div 0\) is not defined.
6. If \(a,b,c\) are integers, then \(\left({a \div b} \right) \div c \ne a \div \left({b \div c} \right),\) unless \(c = 1.\)
Example: \(\left({8 \div 4} \right) \div 2 = 1,\) but \(8 \div \left({4 \div 2} \right) \ne 1\)
7. If \(a,b,c\) are non-zero integers and \(a > b,\) then
\(\left({a \div c} \right) > \left({b \div c} \right),\) if \(c\) is positive
\(\left({a \div c} \right) < \left({b \div c} \right),\) if \(c\) is negative

In the same way, we have properties of division for natural numbers and whole numbers; we have some properties associated with the division of integers. Let us learn them in detail.

Closure Property of Division of Integers

If a mathematical operation is closed for a number system, it means that the result will also belong to the same number system when we do the operation.

For example, if \(p\) and \(q\) are two rational numbers, then if \(p + q\) is also a rational number, the addition is said to be closed under rational numbers.
Similarly, if \(a\) and \(b\) are integers, and \(a \div b\) is an integer, then it is said that division is closed under integers. But, when integers are considered, \(a \div b\) is not necessarily an integer.
Example: \(15 \div 4 = 3.75\) is not an integer but a rational number.

Therefore, closure property does not hold good for the division or division is not closed under integers.

Commutative Property of Integers Under Division

If the order of the operands doesn’t influence the result of an operation, then that operation is said to be commutative.

For example, if \(x\) and \(y\) are two rational numbers, and if \(x + y = y + x,\) then the addition is commutative in rational numbers.
But, the division of integers is not commutative. In other words, if \(a\) and \(b\) are two integers, then \(a \div b \ne b \div a\)
Example: Consider \(16 \div 8\) and \(8 \div 16\)
\(16 \div 8\, = 2\,\) and \(8 \div 16\, = \frac{1}{2}\)
We can clearly see that \(16 \div 8\, \ne \,8 \div 16.\)
So, the division is not commutative in integers.

Associative Property of Division of Integers

If the grouping of operands doesn’t change the result, then the operation is said to be associative. For example, for any three rational numbers \(x,y\) and \(z,\) if \(x + \left({y + z} \right) = \left({x + y} \right) + z,\) then the addition is associative in integers.
But, the division of integers is not associative. In other words, if \(a,b,c\) are integers, then \(\left({a \div b} \right) \div c \ne a \div \left({b \div c} \right)\)
Example: Consider three integers \(24,12\,\& \,2.\)
\(\left({24 \div 12} \right) \div 2 = 2 \div 2 = 1\)
\(24 \div \left({12 \div 2} \right) = 24 \div 6 = 4\)
\( \Rightarrow \left({24 \div 12} \right) \div 2 \ne 24 \div \left({12 \div 2} \right)\)
That is, the division is not associative in integers.

Solved Examples

Q.1. Divide:
(i) 84 by 7
(ii) -91 by 13
(iii) -98 by -14
(iv) 324 by -27
Ans: (i) We have, \(84 \div 7 = \frac{{84}}{7} = 12\)
(ii) We have \( – 91 \div 13 = – \frac{{91}}{{13}} = – 7\)
(iii) We have \( – 98 \div \left({ – 14} \right) = \frac{{ – 98}}{{ – 14}} = 7\)
(iv) We have \(324 \div \left({ – 27} \right) = \frac{{324}}{{ – 27}} = – 12\)

Q.2. Find the quotient in each of the following:
(i) \(\left({ – 1728} \right) \div 12\)
(ii) \(\left({ – 15625} \right) \div \left({ – 125} \right)\)
(iii) \(30000 \div \left({ – 100} \right)\)
Ans: (i) We have \(\left({ – 1728} \right) \div 12 = \frac{{ – 1728}}{{12}} = – 144\)
(ii)We have \(\left({ – 15625} \right) \div \left({ – 125} \right) = \frac{{ – 15625}}{{ – 125}} = \frac{{15625}}{{125}} = 125\)
(iii) We have \(30000 \div \left({ – 100}\right) = \frac{{30000}}{{ – 100}} = – \frac{{30000}}{{100}} = – 300\)

Q.3. Find the value of
(i) \(\left[{32 + 2 \times 17 + \left({ – 6} \right)}\right] \div 15\)
(ii) \(\left({17 + 17}\right) \div \left({25 – 42}\right)\)
Ans: We have \(\left[{32 + 2 \times 17 + \left({ – 6} \right)}\right] \div 15\)
\( = \left[{32 + 34 + \left({ – 6} \right)} \right] \div 15\)
\( = \left({66 – 6} \right) \div 15\)
\( = 60 \div 15\)
\( = \frac{{60}}{{15}} = 4\)
Therefore, \(\left[{32 + 2 \times 17 + \left({ – 6} \right)} \right] \div 15 = 4\)
(ii) We have \(\left({17 + 17} \right) \div \left({25 – 42} \right)\)
\( = \left({34} \right) \div \left({ – 17} \right)\)
\( = \frac{{34}}{{ – 17}} = – \frac{{34}}{{17}} = – 2\)
Therefore, \(\left({17 + 17} \right) \div \left({25 – 42} \right) = – 2\)

Q.4. Simplify \(\left\{{36 \div \left({ – 9} \right)} \right\} \div \left\{{\left({ – 24} \right) \div 6} \right\}\)
Ans: We have \(\left\{{36 \div \left({ – 9} \right)} \right\} \div \left\{{\left({ – 24} \right) \div 6} \right\}\)
\( = \frac{{36}}{{ – 9}} \div \frac{{ – 24}}{6}\)
\( = – 4 \div – 4\)
\( = \frac{{ – 4}}{{ – 4}}\)
\( = 1\)
Therefore,\(\left\{{36 \div \left({ – 9} \right)} \right\} \div \left\{{\left({ – 24} \right) \div 6} \right\}\)

Q.5. In a test (+5) marks are given for every correct answer and (-2) marks are given for every incorrect answer, (i) Radhika answered all questions and scored 30 marks though she got 10 correct answers, (ii) Jay also answered all the questions and scored (-12) marks though he got 4 correct answers. How many incorrect answers had they attempted?
Ans: (i) Marks are given for one correct answer \( = 5\)
So, marks allotted for \(10\) correct answers \( = 5 \times 10 = 50\)
Radhika’s score \( = 30\)
Marks obtained for incorrect answers \( = 30 – 50 = – 20\)
Marks are given for one wrong answer \( = – 2\)
Therefore, number of incorrect answers \( = \left({ – 20} \right) \div \left({ – 2} \right) = 10\)
(ii) Marks given for \(4\) correct answers \( = 5 \times 4 = 20\)
Jay’s score \( = – 12\)
Marks obtained for incorrect answers \( = – 12 – 20 = – 32\)
Marks are given for one wrong answer \( = – 2\)
Therefore, number of incorrect answers \( = \left({ – 32} \right) \div \left({ – 2} \right) = 16\)

Summary

This article discussed the definition of integers, the meaning of addition, subtraction, and multiplication of integers. Also, we have learnt the division of integers and their properties and solved some example problems on the same.

Frequently Asked Questions

We have provided some frequently asked questions about Properties of Division of Integers here:

Q.1. What is the associative property of the division of integers?
Ans: If the grouping of operands doesn’t change the result, then the operation is said to be associative. The division of integers is not associative. In other words, if \(a,b,c\) are integers, then \(\left({a \div b} \right) \div c \ne a \div \left( {b \div c} \right).\)

Q.2. What is the closure property of the division of integers?
Ans: If a mathematical operation is closed for a number system, it means that the result will also belong to the same number system when we do the operation.
If \(a\) and \(b\) are integers, then \(a \div b\) is not necessarily an integer. Therefore division is not closed under integers.

Q.3. Write the value of \(5 \div 0.\)
Ans: We know that if \(a\) is an integer, then \(a \div 0\) is not defined. So, \(5 \div 0\) is not defined.

Q.4. What are the properties of the division of integers?
Ans: 1. If \(a\) and \(b\) are integers, then \(a \div b\) is not necessarily an integer.
2. If \(a\) is an integer different from \(0,\) then \(a \div a = 1.\)
3. For every integer \(a,\) we have \(a \div 1 = a.\)
4. If \(a\) is a non-zero integer, then \(0 \div a = 0\)
5. If \(a\) is an integer, then \(a \div 0\) is not defined.
6. If \(a,b,c\) are integers, then \(\left({a \div b}\right) \div c \ne a \div \left({b \div c} \right),\) unless \(c = 1.\)

Q.5. What is the commutative property of the division of integers?
Ans: If the order of the operands doesn’t influence the result of an operation, then that operation is said to be commutative.
The division of integers is not commutative. In other words, if \(a\) and \(b\) are two integers, then \(a \div b \ne b \div a.\)

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