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$$21 \equiv 4\pmod{17} \equiv \ ?$$

What does this expression mean? I know what mod mean, and what $\equiv$ means, but together this equation doesn't seems meaning anything. The $21\equiv 4 \mod{17}$ makes sense, but what's the succeeding part ?

amWhy
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Yo Yo
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    What context did you see that in? – John Dvorak Mar 09 '20 at 16:11
  • it's just a quiz with this one sentence, and the answer is 38 – Yo Yo Mar 09 '20 at 16:12
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    There are many integers which are $4\pmod {17}$. $21$ and $38$ are certainly examples, but there are infinitely many examples. Without more information there's no way to say why $38$ was the official solution and not, say, $55$. – lulu Mar 09 '20 at 16:15
  • You have the solution set ${...-27, -13, 4, 21, 38, ...}$. – amWhy Mar 09 '20 at 16:21
  • Sometime people who write quizzes make mistakes. It's not clear what they are asking and putting the $\mod 17$ in the middle is confusing but not technically wrong. I would have to assume (but wouldn't be able to say it is obvious) that they want us to name another integer that is congruent to $4$ or $21\pmod{17}$. I'd personally answer $-13$ because... well, why not? As lulu points out there are infinitely many answers. I think this question was written by someone who knew enough to know what $\mod 17$ means didn't know enough to know there are infinite answers. Life is irritating. – fleablood Mar 09 '20 at 16:25
  • The question was very poorly posed to you. It should read: $21 \equiv 4\pmod{17}$. Find the next least greater integer $n$ such that $n\equiv 4 \pmod{17}$. – amWhy Mar 09 '20 at 16:40
  • Thanks, now it makes sense. – Yo Yo Mar 09 '20 at 16:44
  • Glad to help, @YoYo – amWhy Mar 09 '20 at 16:48
  • " Find the next least greater integer n such that n≡4(mod17)." If that is what the question was going for. Because the question was so very badly written I don't think we can ever know what they did have in mind. Anyhow, I think the OP understands the subject well enough she should not worry. – fleablood Mar 09 '20 at 16:53

1 Answers1

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$21\equiv4\bmod17$ means $17|21-4$. In fact, $21\equiv4+17k\bmod17$ for all $k\in \mathbb Z$.

In particular, taking $k=2$, we have $21\equiv38\bmod17$.

So $38$ is an answer, but there are infinitely many acceptable others, including $55$ for example.

J. W. Tanner
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