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Why is $34x = 50 \text{ mod } 33 \Leftrightarrow 1x = 17 \text{ mod } 33$?

I have found that task on this site: Find all solutions; $17x \equiv 25 (\text{ mod } 33)$

Looking at Bill Dubuque's answer I don't see why this is the case, how to know that so fast?

I hope someone can give an explanation so I can understand it?

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    Because $34\equiv 1$ and $50\equiv 17\mod 33$. – 2017-02-12
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    @Bernard Oh thanks Bernard, understood :) – 2017-02-12

1 Answers 1

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We have:

$34 ≡ 1 \quad(mod 33)$

$ 50 ≡ 17\quad (mod 33)$

Using the fact that if $a ≡ b$ and $c ≡ d \quad$ (mod m), then $ac ≡ bd \quad $(mod m), we find that:

$34x ≡ x \quad (mod 33)$

We conclude that $x ≡ 17$ (mod 33)