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$$36x \equiv 81 \pmod{21}$$

This is what I got, I made it in another way, but I have to do this with Fermat's little theorem.

This is what I made:

$$\begin{align}36x &\equiv 81 \pmod{21}\\ 12x &\equiv 27 \pmod 7\\ \gcd(12,7)&=1\end{align}$$

$\varphi(7)=6$

$$x ≡ 27 * 12 ^{\varphi(7)-1}\equiv 6\cdot(-2)^5 \equiv -(-2)^5 = 25 = 32 ≡ 4 \pmod 7$$

$$x \in \{4, 11, 18\}$$

And this is where I got stuck with Fermat's little theorem:

$$\begin{align}36x &\equiv 81 \pmod{21}\\ 6^2 &\equiv 9^2 \pmod{21}\end{align}$$

Now what should I do? I was looking for examples, but unfortunately I didn’t understand.

Blue
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Gsomeone
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    You prove that all solutions of the equation are among $x\in{4,11,18}$, and you can easily verifiy that all these indeed satisfy the equation. What is bothering you? – Pomponazzo May 04 '21 at 07:57
  • Notice I edited your question to improve the formatting. It is strongly advised that you use Mathjax to format your questions on this site - it's like LaTeX for the web. I edited your question this time since you are new, but in future, please format the question yourself. See here for a quick guide: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – 5xum May 04 '21 at 07:59
  • Unfortunately, I have to solve and derive with Fermat's little theorem and not with the way I did it – Gsomeone May 04 '21 at 08:04
  • Thanks for the formatting! – Gsomeone May 04 '21 at 08:04
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    You can't directly apply Fermat (or Euler) to invert $36$ since it is not coprime to the modulus $21$. Cancelling $,3 = \gcd(36,21),$ is the correct way to proceed. See here for the general method. – Bill Dubuque May 04 '21 at 09:17

1 Answers1

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By CRT, we are reduced to computing modulo $3,7$.

Modulo $7$, you can divide out $9$ to start, getting $4x\equiv9\pmod 7 $, since $(7,9)=1$.

We have $36x\equiv81\pmod3\iff 0\equiv0$. That is, every $x $ is a solution.

$\varphi (7)=6$. We apply Fermat's little theorem.

Get that $4^{-1}\equiv4^5\equiv (-3)^5\equiv-5\equiv2\pmod7$.

So we get $x\equiv18\equiv4\pmod7$.

There's not a way to solve the problem solely with Fermat's little theorem w/o cancellation.

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    9 is not invertible mod 21. – Jaap Scherphuis May 04 '21 at 08:53
  • oh no. I need to do it over. @JaapScherphuis –  May 04 '21 at 08:56
  • The completely new revision has a big gap (for beginners), viz. you don't explain how "dividing out $9$" changes the modulus from $21$ to $7$, nor do you say whether that is a unidirectional or bidirectional inferences. But OP already said they cannot use this method, – Bill Dubuque May 04 '21 at 09:24
  • Thanks for the input @BillDubuque CRT reduces it to mod $3,7$ calculation. But mod $3$ we have a tautology. OP wanted to use Fermat's little theorem. Did that when inverting $4$, though it might have been easier to do it "mentally". I suspect the OP is confused about what he/she can do. –  May 04 '21 at 09:47
  • Your first answer essentially divided by zero, Your latest one does another mystical division. Even I can't figure out what you intend from that prior comment, so how can a beginner? Please try to be more precise. – Bill Dubuque May 04 '21 at 09:54
  • @BillDubuque all I'm saying is that the OP wants to use Fermat's little theorem, and that was done. Also, since $9$ is invertible $\pmod7$, the reduction is bi-directional. –  May 04 '21 at 09:59
  • Please explain in the answer. Again, OP already solved it that way, and they said that is not allowed. It seems they want to apply Fermat/Euler w/o cancellation, i.e. mod $21$, which is not possible, as I commented on the question. – Bill Dubuque May 04 '21 at 10:01
  • @BillDubuque I will edit to that effect. –  May 04 '21 at 10:05