Question

If $\mathit{u}\mathbf{\equiv }\mathit{v}\left(modn\right)$and$\mathit{v}$ is a solution of $\mathbf{6}\mathit{x}\mathbf{+}\mathbf{5}\mathbf{\equiv }\mathbf{7}\left(modn\right)$, then show that$\mathit{v}$ is also a solution.[Hint: Theorem 2.2]

Short answer

The given congruence equation is proved using the property of congruence by consideringas the solution of the given equation and then substituting the obtained value in the given equation, which at last implies that v is also a solution of the given equation.

Step-by-step solution

Define congruence.

Congruence: An integer is said to be congruent to $b$ modulo $n$ , where the number $b$ is an integer and $n$ is a positive integer, if $n$ divides the expression $\left(a-b\right)$.

Let the solution of the given equation be u.

The given congruence relation is

$6x+5\equiv 7\left(\mathrm{mod}n\right)$ …(1)

If the solution of equation (1) is u, then it satisfies the congruence equation

$u\equiv v\left(\mathrm{mod}n\right)$ …(2)

This implies that,

$\begin{array}{c}u-v=pn\\ u=v+pn\end{array}$ …(3)

Where, $p$ is an integer.

Substitute the value of u in equation (1) and obtain an equation which satisfies the property of congruence.

Equation (1) can be re-written as, since equation (2) is a solution of equation (1)

$6u+5\equiv 7\left(\mathrm{mod}n\right)$

The above equation is converted as,

$6x+5-7\equiv qn$…… (4)

Where $\mathit{q}$ is an integer.

Substitute$v+pn$for$u$in equation (4) and simplify.

$\begin{array}{c}6\left(v+pn\right)+5-7=qn\\ 6v+6pn-2=qn\\ 6v-2=\left(q-6p\right)n\end{array}$

The expression$\left(q-6p\right)$is also an integer. Since the variables$p$and$q$are integers. Therefore, the obtained equation implies that$v$is also a solution of the given congruence equation.

Hence, the required result is proved.