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Given $5^b\equiv 2345[2^{12}]$ for some positive integer $b$. Also, given order of $2345$ modulo $2^{12}$ is $2^9$ and order of $5$ modulo $2^{12}$ is $2^{10}$. We are to find $b$.

What I tried is the following. We know if $a$ be a group element of finite order then $|a^d|=\frac{|a|}{gcd(|a|,d)}$. According to this, $$2^9=|2345|=|5^b|=\frac{|5|}{\gcd(|5|,b)}=\frac{2^{10}}{\gcd(2^{10},b)}$$ which gives $b=2$ only. But this is absurd, since $5^2\not\equiv 2345[2^{12}]$. Where did I do wrong ?

KON3
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    Why does your displayed equation give $b=2$ "only"? As far as I can see, all it gives you is $b\equiv 2\pmod 4$. – Troposphere Aug 18 '21 at 19:22

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You have computed that the order of $5^b=5^2$ is $2^9$. This is correct. Also the order of $2345$ is $2^9$. But this doesn't imply that $5^2$ and $2345$ are equivalent modulo $2^{12}$. In fact, they are not, as you have stated correctly. So your implicit assumption that elements of equal order are equal in the group is not true.

Dietrich Burde
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  • I think the OP's implicit assumption was that $5^b\equiv 2345\bmod{2^{12}}$ has some solution (after all, that was "given" in the text of the exercise). Furthermore he thinks he has proved that $|5^b| = |2345| \implies b=2$, which would would contradict that assumption -- and for this contradiction he only needs the unproblematic $5^b \equiv 2345 \implies |5^b|=|2345|$. The real problem is that his argument for $\implies b=2$ is faulty. – Troposphere Aug 18 '21 at 19:32
  • Yes, you are right. He only shows that $5^2$ has order $2^9$. Still, it seems to me, that he wants to conclude from $|5^b|=|2345|$, that $5^b=2345$ in $C_{2^{12}}$. Anyway, the argument is not sufficient. – Dietrich Burde Aug 18 '21 at 19:43
  • We're playing fill-in-the-blanks with a not extremely clear question here. :-) My proposal is that he thinks he has proved "if $b$ is a solution then $b=2$", whereas your proposal is that he thinks he has proved "$2$ is a solution". (And we both agree that he has actually he has proved neither statement, for separate reasons ...) – Troposphere Aug 18 '21 at 19:48
  • I agree, and your point of view seems even more likely. I don't exactly understand the conclusion "but this is absurd". – Dietrich Burde Aug 18 '21 at 19:57
  • To all. First accept my sincere apology due to late response. Also, thanks to @DietrichBurde as his explanation clearly showed the fault in my argument and also the mistake i made. All I was trying to see, if any how I can derive the value of $b$ from the given information and that is why, I tried to approach in the above format. However you all have clearly showed me through your discussion what was wrong. Thank you all once again. – KON3 Aug 18 '21 at 20:32