Step 1 Given:

\(17x \equiv 3(mod\ 210)\)

Therefore,

\(x\equiv 17^{-1}*3(mod\ 210)\)

Find: \(17^{-1}(mod\ 210)\)

\(210=17\times 12+6\)

\(17=6\times 2+5\)

\(6=5\times 1+1\)

Trace the steps backward.

\(1=6-5\)

\(=6-(17-6*2)\)

\(=3*6-17\)

\(=3(210-17*12)-17\)

\(3*210-36*17-17\)

\(3*210-37*17\)

Thus, \(17^{-1}(mod\ 210)=-37(mod\ 210)=173(mod\ 210)\)

Step 2 Hence,

\(x\equiv 17^{-1}*3(mod210)\)

\(\equiv 173*3(mod\ 210)\)

\(\equiv 519(mod\ 210)\)

\(\equiv 99(mod\ 210)\)

Thus, \(x\equiv 99(mod\ 210)\)

Therefore, \(17\times 99=1683\equiv 3(mod\ 210)\)

Step 3 Result:

Thus, \(x\equiv 99(mod\ 210)\)

Therefore, \(17\times 99=1683\equiv 3(mod\ 210)\)

\(17x \equiv 3(mod\ 210)\)

Therefore,

\(x\equiv 17^{-1}*3(mod\ 210)\)

Find: \(17^{-1}(mod\ 210)\)

\(210=17\times 12+6\)

\(17=6\times 2+5\)

\(6=5\times 1+1\)

Trace the steps backward.

\(1=6-5\)

\(=6-(17-6*2)\)

\(=3*6-17\)

\(=3(210-17*12)-17\)

\(3*210-36*17-17\)

\(3*210-37*17\)

Thus, \(17^{-1}(mod\ 210)=-37(mod\ 210)=173(mod\ 210)\)

Step 2 Hence,

\(x\equiv 17^{-1}*3(mod210)\)

\(\equiv 173*3(mod\ 210)\)

\(\equiv 519(mod\ 210)\)

\(\equiv 99(mod\ 210)\)

Thus, \(x\equiv 99(mod\ 210)\)

Therefore, \(17\times 99=1683\equiv 3(mod\ 210)\)

Step 3 Result:

Thus, \(x\equiv 99(mod\ 210)\)

Therefore, \(17\times 99=1683\equiv 3(mod\ 210)\)