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• If the integer (–1) is multiplied odd number of times to itself the product is –1. That is (–1) × (–1)
× (–1) × ... odd number of times = –1

Example 4: Find the product for each of the following:
(a) (–9) × 5 × 3 (b) 3 × (–17) × (–2) (c) (–13) × (–14) × (–1)

(d) 16 × 23 × 15 (e) (–25) × (–4) × (–3) × 5 (f) (–20) × (–4) × (–5) × 8
Solution: (a) (–9) × 5 × 3 = –(9 × 5) × 3 = –45 × 3 = –135

(b) 3 × (–17) × (–2) = –(3 × 17) × (–2) = –51 × (–2) = +(51 × 2) = +102 or 102
(c) (–13) × (–14) × (–1) = + (13 × 14) × (–1) = +182 × (–1) = –182

(d) 16 × 23 × 15 = 16 × 345 = 5520
(e) (–25) × (–4) × (–3) × 5 = +(25 × 4) × (–3) × 5 = 100 × (–3) × 5 = –(100 × 3) × 5 = –1500

(f) (–20) × (–4) × (–5) × 8 = – (20 × 4 × 5 × 8) = –(3200) = –3200

Think and Answer
The temperature of a place drops by 3°C per hour for 5 hours at night. Which expression
does not describe the change in temperature in the particular period?
(a) (–3) + (–3) + (–3) + (–3) + (–3) (b) (–3) × 5
(c) (–3) × (–3) × (–3) × (–3) × (–3) (d) – 3 – 3 – 3 – 3 – 3

Properties of Multiplication of Integers

Closure Property The product of any two integers is again an integer.
If a and b are any two integers, then a × b is also an integer.

E.g., (–5) × 3 = –15, which is an integer.
Thus, integers are closed under multiplication.

Commutative Property Integers hold the commutative property for multiplication, as the
product of two integers in any order is the same.
E.g., Let us take any two integers (–4) and (–3).
(–4) × (–3) = (4 × 3) = 12 [Q –ve integer × –ve integer = +ve integer]
Or (–3) × (–4) = (3 × 4) = 12 [Q –ve integer × –ve integer = +ve integer]
Clearly, (–4) × (–3) = (–3) × (–4) = 12
Hence, if a and b are any two integers, then a × b = b × a.

Associative Property Integers hold the associative property for multiplication, as the product
of any three integers grouped in any order is the same.

E.g., Let us take any three integers (–3), 2 and (–5).
So, (–3) × 2 × (–5) = [(–3) × 2)] × (–5) = (–6) × (–5) = 30
Or (–3) × 2 × (–5) = (–3) × [2 × (–5)] = (–3) × (–10) = 30
Clearly, [(–3) × 2 × (–5)] = (–3) × [2 × (–5)] = 30

Hence, if a, b and c are any three integers, then a × (b × c) = (a × b) × c.

Mathematics-7 14