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Commutative Property Let us observe the following example:
1 1
8 ÷ 4 = 2 but 4 ÷ 8 = ; since, 2 ≠
2 2
Therefore, 8 ÷ 4 ≠ 4 ÷ 8
Thus, integers do not hold the commutative property for division, as
the quotient of two integers when divided in any order is not the same.
If a and b are any two integers, then a ÷ b ≠ b ÷ a

Associative Property Let us take any three integers 36, (–12) and 3.
36 ÷ [(–12) ÷ 3] = 36 ÷ (–4) = –9
And, [36 ÷ (–12)] ÷ 3 = (–3) ÷ 3 = –1
Clearly, 36 ÷ [(–12) ÷ 3] ≠ [36 ÷ (–12)] ÷ 3
Thus, integers do not hold associative property for division.
Hence, if a, b and c are any three integers, then a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c

Property of 1 Any integer divided by 1 gives the same integer.
For example: (a) 7 ÷ 1 = 7 (b) (–9) ÷ 1 = –9
Thus, if a is an integer, then a ÷ 1 = a.

Property of zero Any non-zero integer divided by zero is meaningless (or not defined).
But zero divided by any non-zero integer always gives zero, i.e., for
any integer a, a ÷ 0 is not defined but 0 ÷ a = 0, where a ≠ 0.
Example: (a) 0 ÷ 4 = 0 (b) 0 ÷ 9 = 0 (c) 0 ÷ (–2) = 0

Example 8: Find the quotient.

(a) –51 ÷ 3 (b) 105 ÷ (–5)
(c) 112 ÷ (–8) (d) –140 ÷ (–1)

(e) (–53) ÷ (–53) (f) (+93) ÷ (–31)
Solution: (a) –51 ÷ 3 = –(51 ÷ 3) = –17 (b) 105 ÷ (–5) = –(105 ÷ 5) = –21

(c) 112 ÷ (–8) = –(112 ÷ 8) = –14 (d) (–140) ÷ (–1) = 140 ÷ 1 = 140
(e) (–53) ÷ (–53) = 53 ÷ 53 = 1 (f) (+93) ÷ (–31) = –(93 ÷ 31) = –3

Example 9: The product of two integers is 216. If one of the integers is –12, find the other integer.
Solution: The product of two integers = 216
One of the two integers = –12

Therefore, the required other integer = 216 ÷ (–12) = – (216 ÷ 12) = –18
Thus, the required other integer is –18.

Example 10: In a test, (+5) marks are given for every correct answer and (–2) marks are given for
every incorrect answer.
(a) Radhika answered all the questions and scored 30 marks though she got 10 correct answers.

(b) Jay also answered all the questions and scored (–12) marks though he got 4 correct answers.
How many incorrect answers had they attempted?

19 Integers