Breaking RSA using known root Sorry in advance if this question is silly.
Suppose we want to break RSA, that is find the message $m$ s.t. $c \equiv m^e \pmod n$
Now suppose we know an $x$ s.t. $c \equiv x^e \pmod n$. Could we use that information to somehow find $m$?
Breaking an instance of (textbook) RSA encryption is: given $(n,e)$ and $c$, finding $m$ such that $c\equiv m^e\pmod n$ (meaning by definition: $n$ divides $m^e-c\ $), and $0\le m&lt;n$.

For proper establishment of $(n,e)$, there is a single such $m$ for each integer $c$.

If we are additionally (or instead of $c$) given an integer $x$ such that $c\equiv x^e\pmod n$ (meaning by definition: $n$ divides $x^e-c\ $), then we can compute the desired $m$ as $x\bmod n$. That is the uniquely defined $m$ such that $n$ divides $x-m$ and $0\le m&lt;n$.

For non-negative $x$, we can compute $x\bmod n$ as the remainder of the Euclidean division of $x$ by $n$. For negative $x$, we can use that $x\bmod n$ is $(n-1)-((-x-1)\bmod n)$, where $(-x-1)\bmod n$ is computed as the remainder of the Euclidean division of $-x-1$ by $n$.