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Why has the message $P$ to be relative prime to $n$ in RSA encryption?

This should be fault? \begin{align} C &\equiv P^e \pmod{n} \\ &\equiv 101112^{11111357} \pmod{9998000099} \\ &\equiv 3316546434 \pmod{9998000099} \end{align}

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  • $\begingroup$ Please do not use pictures for critical portions of your post. Pictures may not be legible, cannot be searched and are not view-able to some, such as those who use screen readers. I have edited your question to reflect this principle. It is possible that I unintentionally changed the meaning of your question. Please proofread the question to ensure this has not happened. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Jan 19 '18 at 17:23
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    $\begingroup$ There is no need for the message to be relatively prime to the modulus in RSA. Why do you think this? $\endgroup$ – Matthew Towers Jan 19 '18 at 18:11
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$P$ can be any number $< n$, encryption will work. It will produce a valid ciphertext for which decryption still works.

It's also a very small probability (for realistically sized $n$) that this would happen anyway. If anyone were to randomly generate some $P$ and notice that the gcd of $P$ and $n$ was not $1$, that person would have factored $n$ and broken this RSA-instance.

Your example does have gcd of $P$ and $n$ equal to $1$.

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