I am working on the following exercise on the RSA algorithm:
Suppose that Eve has a magic box which creates decryption exponents for $(n, e)$ for a fixed modulus $n$ and for a large number of different public exponents $e$. Let $n = 225022969$. Eve’s magic box gives her the following three encryption/decryption pairs for $n$: $(e_1, d_1) = (70583995, 4911157)$, $(e_2, d_2) = (173111957, 7346999)$, $(e_3, d_3) = (180311381, 29597249)$. Use this information to factor $n$.
My train of thought: we should have the following equations:
$e_1 d_1 = k\phi(n) + 1, \quad k \in \mathbb{Z}$
$e_2 d_2 = s\phi(n) + 1, \quad s \in \mathbb{Z}$
$e_3 d_3 = r\phi(n) + 1, \quad r \in \mathbb{Z}$
Therefore, we can guess $\phi(n)$ and then solve a quadratic equation for $p$ and $q$, where $n = pq$, $\phi(n) = (p-1)(q-1)$.
However, when I factorize $n$ in Wolfram Mathematica, I get that $n = 10867 \cdot 20707$, so none of the equations above is satisfied!
Could you please tell me where I am wrong?