Solve the following linear congruence: 17x congruence 3(mod 210)

Solve the following linear congruence: 17x congruence 3(mod 210)

Question
Congruence
asked 2021-02-25
Solve the following linear congruence: \(17x\ congruence\ 3(mod\ 210)\)

Answers (1)

2021-02-26
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, \(17\times 99=1683\equiv 3(mod\ 210)\)
0

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