Properties of Division: The branch of mathematics that involves the addition, subtraction, division, and multiplication of all kinds of real numbers, including integers is called arithmetic operation. We know integers are specific numbers that include negative numbers, positive numbers, and zero, but no fractions. You must be aware that division is the inverse process of multiplication.

One can describe division as the process of repetitive subtraction. It is a method of grouping things in groups equally. For instance, dividing students into several rows in morning assembly in schools. In this article, we will provide you with detailed information on all the properties of division.

Define Division

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

We know that dividing \(20\) by \(5\) means finding an integer that, when multiplied with \(5\) gives us \(20.\) Such an integer is \(4.\)

Therefore, we write \(20 \div 5 = 4\) 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 \(36 \div \left( { – 9} \right) = \, – 4\) or, \(\frac{{36}}{{ – 9}} = \, – 4\)

Dividing \(\left( { – 35} \right)\) by \(\left( { – 7} \right)\) means what integer should be multiplied with \(\left( { – 7} \right)\) to get \(\left( { – 35} \right).\)
Such an integer is \(5.\)
Therefore, \(\left( { – 35} \right) \div \left( { – 7} \right) = 5\) 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 not wholly divisible by another number, 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 \(2.\)

Division as Repeated Subtraction

Let \(a\) and \(b\) be two whole numbers. Then, dividing the whole number \(a\) by the whole number \(b\) means determining a whole number \(c\) such that when \(b\) is subtracted \(c\) times from \(a,\) we arrive at \(0\) and we write \(a \div b = c.\)
Example: To divide \(20\) by \(5,\) we subtract \(5\) repeatedly from \(20\) till we arrive at \(0.\)
\(20 – 5 = 15 – 5 = 10 – 5 = 5 – 5 = 0\)

What are the Properties of Division?

The properties of the division are given below:

Property 1: If \(a\) and \(b\) (where \(b\) is not equal to zero) are the whole numbers, then \(a \div b\) (expressed as \(\frac{a}{b}\)) is not necessarily the whole number.

In other words, the whole numbers are not closed for the division.

Verification: we know that dividing a whole number \(a\) by a non-zero whole number \(b\) means finding a whole number \(c\) such that \(a = bc.\)

Consider the division of \(14\) by \(3.\) We find that there is no whole number which, when multiplied by \(3\) gives us \(14.\) So, \(14 \div 3\) is not a whole number. Similarly, \(12,\;5,\;9,\;4,\;37,\;6\) etc. are not whole numbers.

Property 2: If \(a\) is any whole number, then \(a \div 1 = a.\)
This means any real number divided by the number \(1\) gives the quotient as the number itself.

Verification: We know that,
\(1 \times 5 = 5\,\,\,\,\,\therefore \,\,\,\,5 \div 1 = 5\)
\(21 \times 1 = 21\,\,\,\,\,\therefore \,\,\,\,21 \div 1 = 21\)
\(0 \times 1 = 0\,\,\,\,\,\therefore \,\,\,\,0 \div 1 = 0\)
\(1 \times 1 = 1\,\,\,\,\,\therefore \,\,\,\,1 \div 1 = 1\) and so on.

Property 3: If \(a\) is any whole number other than zero, then \(a \div a = 1.\)
In this way, you can also say that any whole number (other than zero) divided by itself gives \(1\) as the quotient.

Verification: We know that,
\(12 = 12 \times 1\,\,\,\,\,\,\therefore \,\,\,\,\,\,\,12 \div 12 = 1\)
\(7 = 7 \times 1\,\,\,\,\,\,\therefore \,\,\,\,\,\,\,7 \div 7 = 1\)
\(6 = 6 \times 1\,\,\,\,\,\,\therefore \,\,\,\,\,\,\,6 \div 6 = 1\)
\(1 = 1 \times 1\,\,\,\,\,\,\therefore \,\,\,\,\,\,\,1 \div 1 = 1\) and so on.

Property 4: When zero is divided by any whole number (other than zero) it gives the quotient as the number zero.
In other words, if \(a\) is the whole number other than zero, then \(0 \div a = 0\)

Verification: We have,
\(0 \times 5 = 0\,\,\,\,\,\,\,\therefore \,\,\,\,\,\,0 \div 5 = 0\)
\(0 \times 9 = 0\,\,\,\,\,\,\,\therefore \,\,\,\,\,\,0 \div 9 = 0\)
\(0 \times 1 = 0\,\,\,\,\,\,\,\therefore \,\,\,\,\,\,0 \div 1 = 0\) and so on.

Remark: To divide \(6\) by \(0,\) we must find a whole number which, when multiplied by \(0\) gives us \(6.\) No such number can be obtained; we, therefore, say that division by \(0\) not defined.

Property 5: Let \(a,\;b\) and \(c\) be whole numbers and \(b \ne 0,\;c \ne 0.\) If \(a \div b = c,\;\) then \(b \times c = a.\)

Verification: We have,
\(12 \div 4 = 3\,\,\,\,\,\,\therefore \,\,\,\,\,\,3 \times 4 = 12\)
\(42 \div 7 = 6\,\,\,\,\,\,\therefore \,\,\,\,\,\,7 \times 6 = 42\) and so on. \(\;b \times c = a.\)

Property 6: Let \(a,\;b\) and \(c\) be the whole numbers and \(b \ne 0,\;c \ne 0.\) If \(b \times c = a,\) then \(a \div c = b\) and \(a \div b = c\)

Verification: We have,
\(12 = 3 \times 4\,\,\,\therefore \,\,\,\,12 \div 3 = 4\) and \(12 \div 4 = 3\)
\(4 \times 9 = 36\,\,\,\therefore \,\,\,\,36 \div 4 = 9\) and \(36 \div 9 = 4\)

Property 7: (Division Algorithm) If a whole number \(a\) is divided by a non-zero whole number \(b\) then there exists whole numbers \(q\) and \(r\) such that \(a = bq + r,\) where either \(r = 0\) or, \(r < b.\)

This can be expressed as:

Properties of Division in Integers

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

1. If \(a\) and \(b\) are integers, then \(a \div b\) is not necessarily an integer.
   Example: \(\;14 \div 2 = 7.\) Here, the quotient is an integer.
   But, in \(\;15 \div 4,\) we observe that the quotient is not an integer. Here, the result is \(\frac{{15}}{4} = 3_4^3.\;\) the quotient is \(3;\) the remainder is \(3\)
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 meaningful.
6. If \(a,\;b,\;c\) are integers, then
   \(a > b \Rightarrow a \div c > b \div c,\) if \(c\) is positive.
   \(a > b \Rightarrow a \div c < b \div c,\) if \(c\) is negative.

Properties of Division Facts

Division: For every multiplication fact, we have two division facts.
Example: For the number \(5\) table, the division facts are \(10 \div 5 = 2,\;25 \div 5 = 5\) and \(50 \div 5 = 10\) and \(5 \times 2 = 10,\;2 \times 5 = 10.\)
The given four expressions form a fact family. The fact family for \(5,7\) and \(35\) they can also be written as shown below:
\(7 \times 5 = 35,\;5 \times 7 = 35\) and \(35 \div 7 = 5,\;\;35 \div 5 = 7\)

Solved Examples on Properties of Division

Q.1. Find the number which, when divided by \(46\) gives a quotient \(11\) and remainder \(18.\)
Ans: We have,
Divisor \(=46,\) Quotient \(=11\) and Remainder \(=18.\)
We have to find the dividend. By division algorithm we have,
\({\rm{Dividend}} = {\rm{Divisor}}\, \times \,{\rm{Quotient}}\, + \,{\rm{Remainder}}\)
\(\Rightarrow {\rm{Dividend}} = {\rm{46}}\, \times \,{\rm{11}}\, + \,{\rm{18}}\)
\( = 506 + 18 = 524.\)
Hence the required answer is \(524.\)

Q.2. Find the value of: \(\left[ {32 + 2 \times 17 + \left( { – 6} \right)} \right] \div 15\)
Ans: We have,
\(\left[ {32 + 2 \times 17 + \left( { – 6} \right)} \right] \div 15\)
\(= \left[ {32 + 34 + \left( { – 6} \right)} \right] \div 15 = \left( {66 – 6} \right) \div 15 = 60 \div 15 = \frac{{60}}{{15}} = 4\)
Hence, the required answer is \(4.\)

Q.3. Calculate \(246 \div 246,\,89 \div 89,\,921 \div 921\) using the property of \(a \div a = 1.\)
Ans: We have,
\(246 \div 246,\,89 \div 89,\,921 \div 921\)
We use the property of division \(a \div a = 1\)
So, \(246 \div 246 = 1\)
\(89 \div 89 = 1\)
\(921 \div 921 = 1\)
Hence, the required response is given above.

Q.4. Calculate \(459 \div 1,\;670 \div 1,\;9871 \div 1\) using the property \(a \div 1 = a.\)
Ans: We have,
\(459 \div 1,{\rm{\;}}670 \div 1,{\rm{\;}}9871 \div 1\)
We use the property of division \(a \div 1 = a\)
So, \(459 \div 1 = 459\)
\(670 \div 1 = 670\)
\(9871 \div 1 = 9871\)
Hence, the required response is given above.

Q.5. Calculate \(0 \div 2981,\;0 \div 199,\;0 \div 9991,\) using the property of \(0 \div a = 0.\)
Ans: We have,
\(0 \div 2981,{\rm{\;}}0 \div 199,{\rm{\;}}0 \div 9991\)
We use the property of division \(0 \div a = 0\)
So, \(0 \div 2981 = 0\)
\(0 \div 199 = 0\)
\(0 \div 9991 = 0\)
Hence, the required response is given above.

Summary

Division is defined as the opposite operation of the multiplying of integers. Division is also described as repeated subtraction. There are four significant terms used in division namely, dividend, divisor, quotient and remainder. It is important to note that when you divide any number with the number \(1\) you get the same number. This is known as identifying property of division. In this article, we learned about the properties of the division of integers and learned about the facts on properties of division.

FAQs on Properties of Division

Q.1. What is the division property called?
Ans: The division property of equality states that the two sides remain equal when we divide both sides of an equation by the same non-zero number. If \(a,b\) and \(c\) are real numbers such that \(a = b\) and \(c \ne 0,\) then \(ac = ac.\)

Q.2. What are the three main parts of division?
Ans: The three main parts of the division are the dividend, the divisor and the quotient.

Q.3. What are the properties of division and multiplication?
Ans: The properties of multiplication are Commutativity, Multiplication by Zero, Existence of Multiplication identity, Associativity, Distributivity of Multiplication over Addition.
The properties of the division are 1. If \(a\) and \(b\) (\(b\) is not equal to zero) are whole numbers, then \(a \div b\) (expressed as \(\frac{a}{b}\)) is not necessarily a whole number.
2. Any whole number divided by \(1\) gives the quotient as the number itself.
3. Any whole number (other than zero) divided by itself gives \(1\) as the quotient.
4. Zero divided by any whole number (other than zero) gives the quotient as zero.

Q.4. What is an example of multiplicative identity property?
Ans: In the identity property of multiplication, the product of any number that is multiplied by the number \(1\) is the number itself.
Therefore, \(16 \times 1 = 16\;\) defines the multiplicative identity.

Q.5. Is there an identity property of division?
Ans: When you divide any number with the number \(1\) you get the same number known as identifying property of division.

We hope this detailed article on Properties of Division helps you in your preparation. If you get stuck do let us know in the comments section below and we will get back to you at the earliest.