Question

Solve the system of congruence’s that arises in Example 8.

Short answer

$\begin{array}{r}65.903,070.37+2.912,285.33+51.921,690.24+10.941,094.4=3,397,886,480\\ \equiv 3,397,886,480\\ \equiv 537,140(mod89,403,930)\end{array}$

Step-by-step solution

Step: 1

We have

$\begin{array}{r}{m}_1=99\\ m_2=98\\ m_3=97\\ m_4=95\\ m_1=99\end{array}$

Step: 2

So, $m=\mathrm{99.98.97.95}=89,403,930$

We have

$\begin{array}{r}{M}_1=m/m_1=903,070\\ M_2=m/m_2=912,285\\ M_3=m/m_3=921,690\\ M_4=m/m_4=941,094\end{array}$

Using the Euclidean Algorithm, we compute that

role="math" localid="1668591470626" $\begin{array}{r}{y}_1=37\\ y_2=33\\ y_3=24\\ y_4=4\end{array}$

Are inverses of $M_k$modulo $m_k$for k=1,2,3,4 respectively. It follows that the solution

$\begin{array}{r}65.903,070.37+2.912,285.33+51.921,690.24+10.941,094.4=3,397,886,480\\ =3,397,886,480\\ is\equiv 537,140(mod89,403,930)\end{array}$