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Question Number 205922 by cortano12 last updated on 03/Apr/24

$$x=2\left(mod7\right)$$$$x=3\left(mod4\right)$$$$x=?$$

Answered by Rasheed.Sindhi last updated on 03/Apr/24

$$x=2\left(mod7\right)\wedge x=3\left(mod4\right)$$$$x=7m+2\wedge 7m+2\equiv 3(mod4$$$$\Rightarrow 7m\equiv 1(mod4$$$$\Rightarrow 7m\equiv 1+4\cdot 5=21(mod4$$$$\Rightarrow m\equiv 3$$$$\Rightarrow x=7m+2=7\left(3\right)+2=23$$$$\Rightarrow x=23+lcm(7,4)\times k$$$$\Rightarrow x=23+28k\mathrm{\forall }k\in \mathbb{Z}$$

Answered by A5T last updated on 03/Apr/24

$$x=7k+2\stackrel{4}{\equiv }3\Rightarrow 3k\stackrel{4}{\equiv }9\Rightarrow k\stackrel{4}{\equiv }3\Rightarrow k=4q+3$$$$\Rightarrow x=7(4q+3)+2=28q+23$$

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