Question

By which integer should (-6) be multiplied to get the product 6?

Answer 1

It should be multiplied by —1

Answer 2

Concept

All the integers together satisfy few fundamental properties. In fact, integers are defined as numbers that follow certain set of properties under addition and multiplication. It means if you find some numbers for which two operations namely addition and multiplication exist such that they follow certain rules then, we call those numbers the integers.

Some of the fundamental properties that any integers a, b, and c follow under addition and multiplication are:

• There exists an additive identity element such that a + 0 = a = 0 + a.

• For each integer a there exist an integer, denoted by -a, such that a + (-a) = 0 = -a + a.

• Multiplication is distributive over addition: a × (b + c) = a×b + a×c.

• There exists an element 1, called multiplicative identity such that 1 × a = a = a × 1.

Given

An integer -6.

Find

An integer, n, such that n × (-6) = 6.

Solution

Calculate the integer 0×6?

Since, 0 when added to other number does nothing, 0 when added to itself is also zero i.e. 0 + 0 = 0. So, we can write

       0 × 6 = (0 + 0) × 6

⇒ 0 × 6 = 0×6 + 0 × 6.     (Multiplication is distributive over addition)

0×6 is also an integer so there exist an additive inverse of it, -(0×6). Adding it to both sides of the above equation,

       -(0×6) + 0×6 = -(0×6) + 0×6 + 0×6

⇒     0 = 0 + 0×6           (since a + (-a) = 0)

⇒     0 = 0×6                 (0 + a = 0)

So, we find

         0×6 = 0.              ...(1)

That is, 0×6 is an integer equal to the identity element.

Calculate the integer (-1) × 6?

Calculate

      (1 + (-1)) × 6 = 0 × 6         (1 + (-1) = 0)

⇒ 1 × 6 + (-1) × 6 = 0 × 6      (Multiplication is distributive over addition)

⇒ 6 + (-1) × 6  =  0 × 6         (1 is multiplicative identity element so 1 × a = a)

⇒ 6 + (-1) × 6 = 0             (Using (1))

Add -6 to both sides

       -6 + 6 + (-1) × 6 = -6 + 0

⇒    0 + (-1) × 6 = -6              (Since  -a + a = 0 and 0 + a = a)

⇒    (-1)×6 = -6.      ...(2)        (Again 0 + a = a)

So, when -1 is multiplied to an integer it gives additive inverse of the integer.

Find n for which n × (-6) = 6?

6 and -6 are the additive inverses of each other. So, 6 is the additive inverse of -6.

Using the above result -1 should be multiplied to -6 to obtain 6, which is the additive inverse of -6.

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