Question

If$\mathbf{\text{s}}$, role="math" localid="1659435566500" $\mathit{t}$are solution of the system in exercise$\mathbf{\text{19}}$ , prove that $\mathbf{\text{s}}\mathbf{\equiv }\mathbf{\text{t\hspace{0.17em}(modr)}}$,where$\mathbf{\text{r}}$ is the least common multiple of $\mathit{m}$and$\mathit{n}$.

Short answer

In the below solution, it is proven that $s\equiv t\left(modr\right)$.

Step-by-step solution

Conceptual Introduction

Let $m_1,m_2\mathrm{....}{m}_f$ be pair wise relatively prime positive integers, (it means that $\mathbf{(}{\mathbf{m}}_{\mathbf{i}}\mathbf{,}{\mathbf{m}}_{\mathbf{j}}\mathbf{)}\mathbf{=}\mathbf{1}$, whenever $i\ne j$) . Assume that the $a_1,a_2\mathrm{.....},a_r$, are any integers.

Then the system,

$\begin{array}{l}x\equiv a_1\left(mod{m}_1\right)\\ x\equiv a_2\left(mod{m}_2\right)\\ x\equiv a_3\left(mod{m}_3\right)\\ .\\ .\\ .\\ x\equiv a_r\left(mod{m}_r\right)\end{array}$

has a solution.

 Addition of congruence.

The two congruence equations can be added or subtracted.

$a\equiv b\left(modp\right)$

$c\equiv d\left(modp\right)$

Add and subtract both of the above equation,

$a\pm c\equiv (b\pm d)\left(modp\right)$

Replace the values in the equation

Now write the system of congruence equations given in the text book,

$x\equiv a\left(modb\right)$

$x\equiv b\left(modn\right)$ ……. (1)

Now it is given that the system has two solutions$s$ and $t$

So we can substitute these solutions to the congruence equation,

$s\equiv a\left(modm\right)$……. (2)

$s\equiv b\left(modn\right)$…… (3)

Similarly for $t$

$t\equiv a\left(modm\right)$……. (4)

$t\equiv b\left(modn\right)$…… (5)

Subtract equation (4) from (2) and (5) from (3)

$s-t\equiv 0\left(modm\right)$

$s-t\equiv 0\left(modn\right)$

Hence the above two equation states that the integers $m$ and $n$ divide the expression .

This proves that the lowest common multiple of these two integers also divide this expression.

Therefore $s\equiv t\left(modr\right)$