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Question Number 115674 by naka3546 last updated on 27/Sep/20

$$Solveforx$$$$x\equiv 1mod3$$$$x\equiv 4mod5$$$$x\equiv 6mod7$$

Answered by floor(10²Eta[1]) last updated on 27/Sep/20

$$\mathrm{x}\equiv 1\left(\mathrm{mod}3\right)\Rightarrow \mathrm{x}=3\mathrm{a}+1$$$$3\mathrm{a}+1\equiv 4\left(\mathrm{mod}5\right)\Rightarrow 3\mathrm{a}\equiv 3\left(\mathrm{mod}5\right)\Rightarrow \mathrm{a}\equiv 1\left(\mathrm{mod}5\right)$$$$\mathrm{a}=5\mathrm{b}+1\Rightarrow \mathrm{x}=3(5\mathrm{b}+1)+1=15\mathrm{b}+4$$$$15\mathrm{b}+4\equiv 6\left(\mathrm{mod}7\right)\Rightarrow 15\mathrm{b}\equiv \mathrm{b}\equiv 2\left(\mathrm{mod}7\right)$$$$\mathrm{b}=7\mathrm{c}+2\Rightarrow \mathrm{x}=15(7\mathrm{c}+2)+4$$$$\Rightarrow \mathrm{x}=105\mathrm{c}+34\mathrm{\forall }\mathrm{c}\in \mathbb{Z}$$

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