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# Euler's Theorem

Euler's theorem relate to the remainder of various powers and has applications ranging from modern cryptography to recreational problem-solving.

Mallory discovers many public keys, including the following:

- \(n=70441807\), \(e=3\)
- \(n=10645627\), \(e=17\)
- \(n=63339281\), \(e=65537\)
- \(n=24864431\), \(e=257\)
- \(n=89221291\), \(e=17\)

What number can Mallory discover is the prime factor of one (or more) of these keys, without needing to factor any of them, making use of a vulnerability of RSA?

Knowing which of the following would allow an attacker to efficiently break the RSA encryption?

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