NTRO exam 2023

Question

the solution of the linear congruence  4x  =  5(mod9)?

1.  6 (mod 9)
2. 8 (mod 9)
3. 9(mod 9)
4. 10 (mod 9)

Answer 1

Option b should be true 

Answer 2

4x =  5 (mod 9)    { Pardon me for writing = as the equivalent sign } 

Now as we know that : ax = b( mod ) is the form equation and we have the ( a,b ) are relative co prime to each other . 

4x = 5(mod 9 ) ---→ This mean that both side should leave the remainder same .

and we now that :       -4 = 5(mod 9 ) , bcz          -4 = 9 * (-1) + 5 { as 5 as the remainder when -4 divided by 9}

so now eqn is :   4x = –4  (mod 9 )  

Now ,

4*x = 4 *(-1) (mod 9)    

Now we have common number both side as 4 to divide we follow : ac = bc (mod n ) 

a = b ( mod n /gcd(c,n))

x = -1 (mod 9) 

now -1 = 8 (mod 9 ) so they are equivalent to each other ) 

Option b seems to be correct . 

 

Comments

+1 for mentioning that in general cancellation property does not hold for linear recurrences i.e. $ac \equiv bc \mod n$ does not imply $a \equiv b \mod n.$

For example, $1\times2 \equiv 4 \times 2  \mod 6$ does not imply $1 \equiv 4 \mod 6$

It is true if gcd(c,n)=1.

Now, taking $-4 \equiv 5 \mod 9$ would work here but sometimes you have to think what should I have to take for solving similar questions.

You could use a mechanical way for these type of questions.

$4x \equiv 5 \mod 9$ which means $x \equiv 5\times 4^{-1} \mod 9$

Now, multiplicative inverse of 4 i.e. $4^{-1}$ is $7$ in $\mathbb{Z}_9$ because $4\times 7 \mod 9 = 1$ (You could also Euclid algorithm to calculate multiplicative inverses.)

Another easy way to solve this question is, by option elimination.

Answer 3

4x=5 mod 9

 adding 4 on both side 4x+4=9 mod 9 ,x= -1

8 ( mod 9) gives -1