Question

If 3x $ \equiv $ 5(mod7), then.
$\left( A \right)$ x $ \equiv $ 2(mod7)
$\left( B \right)$ x $ \equiv $ 3(mod7)
$\left( C \right)$ x $ \equiv $ 4(mod7)
$\left( D \right)$ None of these

Answer 1

Hint – In this particular question use the concept that the equation has only one solution if and only if the greatest common divisor of (a, b) is 1, if GCD is any other number say (4) then the equation has 4 solutions, later on in the solution use the concept that write remainder values in place of the coefficient of x and the coefficient of (mod n) when divided by mod value i.e. n, so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given equation:
3x $ \equiv $ 5(mod7)
It is also called modulo 7, 3 is a multiplicative inverse of 5.
Let us consider a general example of this type of function
Let, ax $ \equiv $b (mod n)
Then this equation has only one solution if and only if the greatest common divisor of (a, b) is 1.
GCD numbers – GCD numbers are those numbers which do not have common factors except 1.
In this general equation, a = remainder, b = remainder and n = divisor.
I.e. we write remainder values when the coefficient of x and the coefficient of (mod n) is divided by n.
So on comparing with given equation we have,
a = 3, b = 5 and n = 7
So, first find out the GCD of (a, b) (i.e. (3, 5))
So, first find out the prime factors of 3 and 5.
Prime numbers are those numbers which are only divisible by 1 and itself.
So the factors of 3 are (1, 3).
And the factors of 5 are (1, 5).
So the common factors of (3 and 5) is only 1.
So the number of solutions of the given equation is only 1.
So the given equation is a unique solution case.
Now in the given equation multiply by 5 on both sides so that when we divide by mod value we get least remainder in the LHS side, so we have,
Therefore, 5(3x) $ \equiv $ (5)5(mod7)
Therefore, 15x $ \equiv $ 25(mod7)
Now divide 15 and 25 by 7 and write in place of 15 and 25 remainder values we have,
Therefore when 15 is divided by 7 i.e.$\dfrac{{15}}{7} = 2\dfrac{1}{7}$, we got remainder 1.
And when 25 is divided by 7 i.e.$\dfrac{{25}}{7} = 3\dfrac{4}{7}$, we got remainder 4.
So the solution of the given equation is
1x $ \equiv $ 4 (mod7)
$ \Rightarrow x \equiv 4\left( {\bmod 7} \right)$
So this is the required answer.
Hence option (C) is the correct answer.

Note – Whenever we face such types of questions the key concept we have to remainder is that multiply both side by the value such that when divided by the mod value we get least remainder on the L.H.S side, so multiply by 5 on both sides in the given equation as above and then divide by 7 and note down the remainder values as above then write these remainder values in place of original numbers doing this we get the required solution.