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# Multiplication of Integers - Product of Three Or More Negative Integers

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## Product of Three Or More Negative Integers:

We observed that the product of two negative integers is a positive integer.

 The product of two positive (+ ve) integers is a positive (+ ve) integer. (+ve number) × (+ve number) = (+ve number) The product of one positive (+ ve) and one negative (-ve) integer is a negative integer. (+ve number) × (-ve number) = (-ve number) (-ve number) × (+ve number) = (-ve number) The product of two negative (-ve) integers is a positive (+ve) integer. (-ve number) × (-ve number) = (+ve number)

Let us observe the following examples:

(a) (– 4) × (– 3) = 12

(b) (– 4) × (– 3) × (– 2) = [(– 4) × (– 3)] × (– 2) = 12 × (– 2) = – 24

(c) (– 4) × (– 3) × (– 2) × (– 1) = [(– 4) × (– 3) × (– 2)] × (– 1) = (– 24) × (– 1)

(d) (– 5) × [(– 4) × (– 3) × (– 2) × (– 1)] = (– 5) × 24 = – 120

From the above products we observe that

(a) the product of two negative integers is a positive integer;

(b) the product of three negative integers is a negative integer.

(c) product of four negative integers is a positive integer.

We find that if the number of negative integers in a product is even, then the product is a positive integer; if the number of negative integers in a product is odd, then the product is a negative integer.

## A Special Case:

Consider the following statements and the resultant products:

(– 1) × (– 1) = + 1

(– 1) × (– 1) × (– 1) = – 1

(–1) × (–1) × (–1) × (–1) = +1

(–1) × (–1) × (–1) × (–1) × (–1) = –1

This means that if the integer (–1) is multiplied even number of times, the product is +1 and if the integer (–1) is multiplied an odd number of times, the product is –1. You can check this by making pairs of (–1) in the statement. This is useful in working out products of integer.

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