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

Question

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?

Because $34\equiv 1$ and $50\equiv 17\mod 33$. – Bernard Feb 12 '17 at 21:14 — Bernard, Feb 12 '17 at 21:14
@Bernard Oh thanks Bernard, understood :) – cnmesr Feb 12 '17 at 21:17 — cnmesr, Feb 12 '17 at 21:17

score 2 · Accepted Answer · answered Feb 12 '17 at 21:23

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)

1 Answers1