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

Question
Congruence
Solve the following linear congruence: $$17x\ congruence\ 3(mod\ 210)$$

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

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