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I'd like to prove $$3+4\mathbb Z = \{ x\in \mathbb Z , 3x \equiv 5 [4]\}$$

The $\Rightarrow$ is trivial: $x\in 3+4\mathbb Z \Rightarrow x\equiv3[4] \Rightarrow 3x\equiv5[4]$

But for the reverse way $\Leftarrow$ I get stuck at a certain point: $3x\equiv 5[4] \Rightarrow 3x\equiv 1[4]$

I am stuck at this point..

PS: I've been thinking about dividing $3x\equiv 1[4]$ by $3\equiv 3 [4]$ which I can do because $gcd(3,4)=1$

but then I would have $\frac{1}{3}$ which is difficult to handle.

Bill Dubuque
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niobium
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2 Answers2

1

$3x \equiv 1 [4] \Rightarrow 3x \equiv 1+8 [4]$ which gives $3x \equiv 9 [4]$ and $x \equiv 3 [4]$ since $(3,4)=1$.

geetha290krm
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1

For the reverse, suppose we have

$\tag 1 3x = 4n + 5$

We need to find $m$ such that

$\quad x = 4m + 3$

This is possible if we can show that $x -3$ is divisible by $4$, which would also be true if $3 \cdot (x-3)$ is divisible by $4$. But, using $\text{(1)}$,

$\tag 3 3 (x-3) = 4n + 5-9 = 4n -4 = 4(n-1)$

and so all is good.

CopyPasteIt
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