Question Number 66815 by Rio Michael last updated on 20/Aug/19
$${solve}\:{the}\:{congruence}\:{equation}\: \\ $$$$\:\:\mathrm{6}{x}\:\equiv\:\mathrm{4}\:\left({mod}\:\mathrm{5}\right)\:\:{i}\:{need}\:{help}\:{please}\:{with}\:{some}\:{explanations} \\ $$
Commented by mathmax by abdo last updated on 20/Aug/19
![if we consider congruence modulo 5 we have 6x ≡4 [5] ⇒6^− x^− =4^− but 6^− =5^− +1^− =1^− and 4^− =(−1)^− ⇒ x^− =−1[5] ⇒x+1 =5k ⇒x =5k−1 with k ∈Z .](https://www.tinkutara.com/question/Q66820.png)
$${if}\:{we}\:{consider}\:{congruence}\:{modulo}\:\mathrm{5}\:\:{we}\:{have}\: \\ $$$$\mathrm{6}{x}\:\equiv\mathrm{4}\:\left[\mathrm{5}\right]\:\Rightarrow\overset{−} {\mathrm{6}}\overset{−} {{x}}=\overset{−} {\mathrm{4}}\:\:\:\:{but}\:\overset{−} {\mathrm{6}}=\overset{−} {\mathrm{5}}\:+\overset{−} {\mathrm{1}}=\overset{−} {\mathrm{1}}\:{and}\:\overset{−} {\mathrm{4}}=\left(−\mathrm{1}\overset{−} {\right)}\:\Rightarrow \\ $$$$\overset{−} {{x}}=−\mathrm{1}\left[\mathrm{5}\right]\:\Rightarrow{x}+\mathrm{1}\:=\mathrm{5}{k}\:\:\:\Rightarrow{x}\:=\mathrm{5}{k}−\mathrm{1}\:\:{with}\:{k}\:\in{Z}\:. \\ $$$$ \\ $$
Answered by Rasheed.Sindhi last updated on 20/Aug/19
$$\mathrm{6}{x}\equiv\mathrm{4}+\mathrm{4}×\mathrm{5}=\mathrm{24}\left({mod}\:\mathrm{5}\right) \\ $$$$\:\:{x}\equiv\mathrm{4}\left({mod}\:\mathrm{5}\right) \\ $$
Answered by mr W last updated on 20/Aug/19
$$\mathrm{6}{x}=\mathrm{5}{n}+\mathrm{4}=\mathrm{5}\left({n}−\mathrm{4}\right)+\mathrm{24} \\ $$$${x}=\frac{\mathrm{5}\left({n}−\mathrm{4}\right)}{\mathrm{6}}+\mathrm{4} \\ $$$${with}\:{n}−\mathrm{4}=\mathrm{6}{m} \\ $$$$\Rightarrow{x}=\mathrm{5}{m}+\mathrm{4} \\ $$$${i}.{e}.\:{x}\equiv\mathrm{4}\left({mod}\:\mathrm{5}\right) \\ $$
Commented by Rio Michael last updated on 20/Aug/19
$${i}\:{still}\:{don}'{t}\:{undestand}\:{how}\:{these}\:{equations}\:{are}\:{solved} \\ $$
Commented by mr W last updated on 21/Aug/19
$${don}'{t}\:{you}\:{understand}\:{what} \\ $$$$\mathrm{6}{x}\equiv\mathrm{4}\:\left({mod}\:\mathrm{5}\right)\:{means}\:{or}\:{where}\:{is}\:{your} \\ $$$${problem}? \\ $$
Commented by Rio Michael last updated on 21/Aug/19
$${i}\:{think}\:{so}\:{sir},{because}\:{am}\:{new}\:{to}\:{this}\:{further}\:{mathematics}\:{stuffs} \\ $$
Commented by mr W last updated on 21/Aug/19
$${a}\equiv{b}\:\left({mod}\:{c}\right)\:{means} \\ $$$${when}\:{a}\:{and}\:{b}\:{are}\:{divided}\:{by}\:{c},\:{they} \\ $$$${have}\:{the}\:{same}\:{remainder}. \\ $$$${in}\:{other}\:{word}: \\ $$$${a}={nc}+{b} \\ $$
Commented by Rio Michael last updated on 25/Aug/19
$${thank}\:{you}\:{so}\:{much}\:{sir} \\ $$