Question

explain briefly properties of division of integers

Answer 1

The following properties of division of integers are: 

(i) If x and y are integers, then x ÷ y is not necessarily an integer. 

For example; 16 ÷ 3, -17 ÷ 5 are not integers. 

(ii) If x is an integer different from 0, then x ÷ x = 1. 

(iii) For every integer x, we have x ÷ 1= x. 

(iv) If x is a non-zero integer, then 0 ÷ x = 0. 

(v) If x is an integer, then x ÷ 0 is not meaningful. 

(vi) If x, y, z are non-zero integers, then (x ÷ y) ÷ z ≠ x ÷ (y ÷ z), unless z = 1. 

(vii) If x, y, z are integers, then 

                    (a) x > y ⇒ x ÷ z > y ÷ z, if z is positive.

                    (a) x > y ⇒ x ÷ z < y ÷ z, if z is negative

Answer 2

$\underline{\bold{Properties \:Of\ Division \:On \:Integers}}$


$\textbf{Property I:}$ If ‘a’ and ‘b’ are two integers, such that b ≠ 0, then a ÷ b may or may not be an integer. Integers are not closed under division.

$\textit{Examples:}$
(i) (–5) ÷ (–5) = 1 [ 1 is an integer ]
(ii) (–10) ÷ 2 = –5 [ –5 is an integer ]
(iii) 15 ÷ (–9) = $\frac { 15}{ - 9}$ [ $\frac { 15}{ - 9}$ is not an integer ]

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$\textbf{Property II:}$ If ‘a’ is an integer and
a ≠ 0, then a ÷ (–a) = –1; (–a) ÷ a = –1.

$\textit{Examples:}$
(i) 9 ÷ (–9) = –1
(ii) (–12) ÷ 12 = –1
(iii) 15 ÷ 15 = 1

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$\textbf{Property III:}$ If ‘a’ is an integer and
a ≠ 0, then a ÷ 1 = a; (–a) ÷ 1 = (–a);
a ÷ (–1) = –a.

$\textit{Examples:}$
(i) 16 ÷ 1 = 16
(ii) (–19) ÷ 1 = (–19)
(iii) 23 ÷ (–1) = (–23)

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$\textbf{Property IV:}$ If ‘a’ ’is an integer and
a ≠ 0, then 0 ÷ a = 0 and 0 ÷ (–a) = 0.

$\textit{Examples:}$
(i) 0 ÷ 8 = 0
(ii) 0 ÷ (–8) = 8


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$\textbf{Hope It Helps !!!}$