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Question Number 151265 by liberty last updated on 19/Aug/21

Commented by Tawa11 last updated on 19/Aug/21

$$\mathrm{Weldone}\mathrm{sir}.$$

Commented by liberty last updated on 19/Aug/21

$$5\mathrm{x}\equiv 2\left(\mathrm{mod}3\right)\Rightarrow \mathrm{x}=1+3\mathrm{a}$$$$3\mathrm{x}\equiv 4\left(\mathrm{mod}7\right)\Rightarrow 9\mathrm{a}+3\equiv 4\left(\mathrm{mod}7\right)$$$$\Rightarrow 2\mathrm{a}\equiv 1\left(\mathrm{mod}7\right)\Rightarrow \mathrm{a}=4+7\mathrm{b}$$$$3\mathrm{x}\equiv 7\left(\mathrm{mod}8\right)\Rightarrow 3(1+3(4+7\mathrm{b}))\equiv 7\left(\mathrm{mod}8\right)$$$$\Rightarrow 3(13+21\mathrm{b})\equiv 7\left(\mathrm{mod}8\right)$$$$\Rightarrow 63\mathrm{b}+39\equiv 7\left(\mathrm{mod}8\right)$$$$\Rightarrow 7\mathrm{b}+7\equiv 7\left(\mathrm{mod}8\right)$$$$\Rightarrow 7\mathrm{b}\equiv 0\left(\mathrm{mod}8\right)\Rightarrow \mathrm{b}\equiv 0\left(\mathrm{mod}8\right)$$$$\Rightarrow \mathrm{b}=8\mathrm{c}$$$$\mathrm{then}\mathrm{x}=1+3\mathrm{a}=1+3(4+7\mathrm{b})$$$$\mathrm{x}=21\mathrm{b}+13=21\left(8\mathrm{c}\right)+13$$$$\mathrm{x}=168\mathrm{c}+13$$$$\mathrm{the}\mathrm{smallest}\mathrm{positive}\mathrm{integer}$$$$\mathrm{x}=13.$$

Commented by otchereabdullai@gmail.com last updated on 22/Aug/21

$$\mathrm{nice}$$

Answered by Rasheed.Sindhi last updated on 19/Aug/21

$$5x\equiv 2+3=5\left(mod3\right)$$$$x\equiv 1\left(mod3\right)..................\left(i\right)$$$$3x\equiv 4(mod7$$$$3x\equiv 4+7\times 2=18(mod7$$$$x\equiv 6\left(mod7\right)....................\left(ii\right)$$$$3x\equiv 7\left(mod8\right)$$$$3x\equiv 7+8=15\left(mod8\right)$$$$x\equiv 5\left(mod8\right)....................\left(iii\right)$$$$\left(i\right)\Rightarrow x=1,4,7,10,13,16,19,....$$$$\left(ii\right)\Rightarrow 6,13,20,27,....$$$$\left(iii\right)5,13,21,29,...$$$$x=13$$

Commented by liberty last updated on 19/Aug/21

$$\mathrm{okay}.\mathrm{thanks}$$

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