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A family member of mine is currently taking high school Algebra 2 and is learning about the basics of modular arithmetic. The following question was asked of the students:


Find all integers n that satisfy the following:

$90\equiv 6 \pmod n$


The family member asked me for help, as she was stuck. But I too was stumped. It has been two years since I took Algebra 2, and I barely remember modular arithmetic. I seem to remember that $90\equiv 6 \pmod n$ means that 90 mod n = 6 mod n -- however, I am not sure how that can help me solve the problem.

I know this is a rather elementary question so I apologize if it has already been answered. That said, I have not been able to find an answer so far, as most sources deal with equations where the unknown value is something other than the one "inside" the mod parenthesis.

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    $\begingroup$ $90\equiv6\pmod n$ means $n$ divides $90-6$ $\endgroup$ Sep 23, 2019 at 3:34
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    $\begingroup$ The definition is surely given in their textbook. What does it say? $\endgroup$ Sep 23, 2019 at 3:40
  • $\begingroup$ @JyrkiLahtonen I don't think they have a textbook, alas... $\endgroup$
    – Will
    Sep 23, 2019 at 3:41

1 Answer 1

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$90\equiv6\pmod n$ means $n$ divides $90-6,$ so the solutions for $n$ are the factors of $84.$

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  • $\begingroup$ Thanks! Why does it mean n divides 90 - 6, though? $\endgroup$
    – Will
    Sep 23, 2019 at 3:38
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    $\begingroup$ That's the definition, so $90=qn+r$ and $6=pn+r$ where $0\le r<n$ $\endgroup$ Sep 23, 2019 at 3:41

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