Question

In the group $\left\{ 1,2,3,4,5,6 \right\}$ under multiplication modulo $7,{{2}^{-1}}\times 4$ is equal to
(a) 1
(b) 4
(c) 2
(d) 3

Answer 1

Hint: Modular operator in arithmetic gives the remainder of a number when we divide it. For example, if we apply the modular operator as 5 mod 2, we would get the result as 1, which is the remainder obtained when dividing 5 by 2.

Complete step-by-step answer:

We have a group $\left\{ 1,2,3,4,5,6 \right\}$ which is under multiplication modulo 7, it means that we need to find the remainder when a given number is divided by 7.

We know what modulo means. Modular arithmetic gives the remainder value when it is divided by a number. When we divide two integers we will have an equation that looks like the following

$\dfrac{A}{B}=Q$

Let us take the remainder to be R

A is dividend

B is the divisor

Q is the quotient

And we require only remainder B, then we use modular concept and a modulo operator which is also abbreviated as a mod.

Now here in the question, we need to find ${{2}^{-1}}\times 4$ mod 7.

So first we need to find what is the value of ${{2}^{-1}}$.

Now we have to understand what ${{2}^{-1}}$ means.

It means that whenever we take modulo of ${{2}^{-1}}$ by 7 it gives 1 or we can say when we divide ${{2}^{-1}}$by 7 it gives remainder 1.

So to find value of ${{2}^{-1}}$ we will suppose that ${{2}^{-1}}=b$

${{2}^{-1}}=b$

Now, it means that

$2b\bmod 7=1\bmod 7$ ( since ${{2}^{-1}}=\dfrac{1}{2}$)

It means that when we divide 2b with 7, it gives a remainder 1.

Now we will use the hit and trial method to get the value of b so that when we divide 2b by 7, it gives remainder 1.

And we can observe that when the value of 2b = 8 and when 8 divided by 7 it gives remainder 1.

So, the value of b = 4 and we know that ${{2}^{-1}}$ is equal to b.

So, ${{2}^{-1}}$ is equal to 4.

Now we know that we need to find ${{2}^{-1}}\times 4$ mod 7, so we will substitute the value of ${{2}^{-1}}$ in it.

And now we will have,

$4\times 4$ mod 7

16 mod 7

When 16 is divided by 7, we will get the remainder 2 as 16 = (2 x 7) + 2.

So the correct option is (c).


Note: The possibility of mistake in the question is to consider ${{2}^{-1}}$ as $\dfrac{1}{2}$ and then multiplying it with 4 in the expression ${{2}^{-1}}\times 4$ mod 7. We will get the wrong answer then.