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As per the title, the task is to

Factor the RSA modulus $n = 3844384501$ knowing that $$3117761185^2 \equiv 1 \pmod{n}\text{.}$$

$n$ being an "RSA modulus" means that it is a product of two primes, i.e. $n = pq$. The task is to find these two primes.

I can see that from what we're given, $$ (3117761185 - 1)(3117761185 + 1) \equiv 0 \pmod{n}\text{,} $$ but I do not know how to proceed from there on. Any hints would be greatly appreciated.

N.B. It would seem that computing $\gcd(3117761184, n)$ and $\gcd(3117761186, n)$ would give me $p$ and $q$ but I have no idea why that would be justified.

Jyrki Lahtonen
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d125q
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    Well.. from the congruence you wrote up we can conclude that $p$ and $q$ are among the union of prime divisors of 3117761184 and 3117761186. – Berci Dec 27 '15 at 15:19
  • You have found $a, b$ such that $ab = kn$ for some $k \in \mathbb{N}$ you don't care about. – Ruben Dec 27 '15 at 15:20
  • @Berci: Why can we conclude that? All I am able to conclude is that $p q \mid 3117761184 \cdot 3117761186$; my background in number theory is probably lacking. – d125q Dec 27 '15 at 15:29
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    @d125q Since $p$ and $q$ are primes, each of them divides either $3117761184$ or $3117761186$. – rogerl Dec 27 '15 at 15:31
  • @rogerl: Thanks a lot. Would it be fine to say that given $a, b$ s.t. $\gcd(a, b) = 1$, we have that $ab \mid c \Leftrightarrow a \mid c \wedge b \mid c$? – d125q Dec 27 '15 at 15:39
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    @d125q Yes, that is correct as well. – rogerl Dec 27 '15 at 15:41

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