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I began by setting up a system of linear equations: $$14\equiv 8a+b \pmod{27}$$ $$5\equiv 26a+b\pmod{27}$$ and then subtracted them to get: $9\equiv 9a \pmod{27}$. I know $9$ doesn't have a multiplicative inverse modulo $27$ but $a=1$ would solve this. I'm afraid I must be making some mistake somewhere, however, as solving this all the way through gives me $b=6$ and then translating the message doesn't quite make sense.

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    $\begingroup$ You missed several solutions. $\endgroup$ – Randall Feb 24 at 20:08
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$$9a\equiv 9 \mod{27}$$ $$a \equiv1 \mod3$$ $$\therefore a=1+3k$$ $$b\equiv14-8a \mod{27}$$ $$b=14-8(1+3k)+27m$$ $$\therefore b=6-24k+27m$$ Where $k,m\in\mathbb{Z}$.

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  • $\begingroup$ So $(a,b)\in \{(1,6),(4,9),(7,12),(10,15),(13,18),(16,21),(19,24),(22,0),(25,3)\}$. Only the ones with invertible $a$, namely $a=1,7,19,25$ warrant further investigation. The others would be uninvertible. $\endgroup$ – Henno Brandsma Feb 24 at 23:07
  • $\begingroup$ They are all invertible mod 27 since its only prime divisor is 3 right? $\endgroup$ – joseph Feb 25 at 1:33
  • $\begingroup$ @josephF yes, sorry, I switched to modulo $26$ in my head again (which is also done sometimes in some texts). $\endgroup$ – Henno Brandsma Feb 25 at 21:58
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$9\equiv 9a \ \pmod{27} \Rightarrow 9(1-a)\equiv 0\ \pmod{27} \Rightarrow 1-a\equiv 0 \pmod{3} \Rightarrow a=1+3k,\ k\in\mathbb{Z}$

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  • $\begingroup$ okay so $a=1,4,7,10,13,16,19,22$ or $25$? $\endgroup$ – joseph Feb 24 at 20:17
  • $\begingroup$ No, I never said that. I was talking about the answer, not your comment. Whether anyone writes $=$ or $\equiv$ hardly matters as long as the context is clear. $\endgroup$ – Randall Feb 24 at 20:20
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    $\begingroup$ @Randall You're right! My mistake, I'll try to fix it $\endgroup$ – giannispapav Feb 24 at 20:29
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    $\begingroup$ Yes, that's correct now. I removed my downvote. $\endgroup$ – Randall Feb 24 at 20:35
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    $\begingroup$ Use \pmod{27} for $\pmod{27}$ ("parenthesized mod" I suppose) for nicer results. $\endgroup$ – Henno Brandsma Feb 24 at 22:33

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