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Question Number 178747 by cortano1 last updated on 21/Oct/22

$$\{\begin{array}{l}3\mathrm{x}=2\left(\mathrm{mod}5\right)\\ 3\mathrm{x}=4\left(\mathrm{mod}7\right)\\ 3\mathrm{x}=6\left(\mathrm{mod}11\right)\end{array}$$$$\mathrm{x}=?$$

Answered by Rasheed.Sindhi last updated on 21/Oct/22

$$\{\begin{array}{l}3\mathrm{x}=2\left(\mathrm{mod}5\right)....\left(\mathrm{i}\right)\\ 3\mathrm{x}=4\left(\mathrm{mod}7\right)....\left(\mathrm{ii}\right)\\ 3\mathrm{x}=6\left(\mathrm{mod}11\right)....\left(\mathrm{iii}\right)\end{array};\mathrm{x}=?$$$$\left(i\right)\Rightarrow 3x\equiv 2+2\times 5\left(mod5\right)$$$$\Rightarrow x\equiv 4\left(mod5\right)$$$$x=5k+4$$$$\left(ii\right)\Rightarrow 3(5k+4)\equiv 4\left(mod7\right)$$$$15k+12\equiv 4\left(mod7\right)$$$$15k\equiv -8+7\left(14\right)\left(mod7\right)$$$$k\equiv 6\left(mod7\right)$$$$k=7l+6$$$$x=5(7l+6)+4=35l+34$$$$\left(iii\right)\Rightarrow 3x\equiv 6\left(mod11\right)$$$$x\equiv 2\left(mod11\right)$$$$35l+34\equiv 2\left(mod11\right)$$$$35l\equiv -32\left(mod11\right)$$$$35l\equiv -32+22\left(11\right)\left(mod11\right)$$$$35l\equiv 210\left(mod11\right)$$$$l\equiv 6\left(mod11\right)$$$$l=11m+6$$$$x=35l+34=35(11m+6)+34$$$$=385m+210+34=385m+244$$$$x=244,385m+244$$

Commented by Tawa11 last updated on 21/Oct/22

$$\mathrm{Great}\mathrm{sir}$$

Commented by cortano1 last updated on 21/Oct/22

$$\mathrm{nice}.\mathrm{thanks}\mathrm{sir}$$

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