Question

Question:- If (a, n) = danddxd,show that $\text{ax}\equiv \text{b(modn)}$has no solution.

Short answer

Answer:-

The value has no solution

Step-by-step solution

Conceptual Introduction

Let $m_1, m_2 ....m_f$ be pair wise relatively prime positive integers, (it means that $\left(m_i,m_j\right)=1$ whenever $i\ne j$) . Assume that the $a_1, a_2....., a_r$, are any integers.

Then the system,

$$\begin{gathered} x\equiv a_1\left(mod m_1\right) \\ x\equiv a_2\left(mod{m}_2\right) \\ x\equiv a_3\left(mod{m}_3\right) \\ . \\ . \\ . \\ x\equiv a_r\left(mod{m}_r\right) \end{gathered}$$

has a solution.

The congruent equation can be written as an algebraic equation.

The congruent equation $x\equiv b(mod n)$ is rewritten as an algebraic equation, with congruent replaced by an equal to sign.

$$\begin{gathered} x\equiv b(mod n) \\ ax-ny=b \end{gathered}$$

The greatest common divisor is d.

If the value is:

.$\left(a,n\right)=d$

Then,

.$d=gcd\left(a,n\right)$

Let d be the divisor dividing both a and n .

Let the value be:

$$\begin{gathered} a=dr \\ n=ds \end{gathered}$$

Where r,s are the elements of Z.

Let be the solutions if the equation is solvable.

Substitute the values as:

$$\begin{gathered} a=dr \\ n=ds \\ x=x_0 \\ y=y_0 \end{gathered}$$

Then, we get:

$$\begin{gathered} a-n{y}_0=b \\ dr{x}_0-ds{y}_0=b \\ d\left(r{x}_0-s{y}_0\right)=b \end{gathered}$$

The value $\left(r{x}_0-s{y}_0\right)$represents an integer hence the equation is simplified asd/b if the condition db is satisfied only the equation will have a solution, whereas in the question it is dxb hence the equation does not have any solution.

Hence, the congruent equation does not have any solution as dxb .