How can I find the prime numbers used in RSA?
Well, this is semi-easy. You should know that public key consists of modulus $N = pq$ with public exponent $e$ and private key is the same modulus with private exponent $d$, where $de=1\pmod{\varphi(N)}$. Now, calculate $k = de-1$ then brute force $gcd(g^{k/2^x}\pm 1, N)$ for random $g$ and small $x$, and with high probability that quickly provides one of the primes. Code example in Python:

&gt;&gt;&gt; n = 0xffffffa95256a837568a41c265f4fe27110814aae19f144762d5cc0bcb931807
&gt;&gt;&gt; e = 0x11
&gt;&gt;&gt; d = 0xc3c3c3817b3335577e69b9d0e48e2bc1fdf71f1f4f73a38a7d628d39739bbaf1
&gt;&gt;&gt; from fractions import gcd
&gt;&gt;&gt; for g in range(1, 9): print (g, gcd(n, pow(g, ((d * e - 1) / 2 ** 2), n) - 1))
...
(1, 115792086900472091959223405310199004229068084504756221892949117819703713273863L)
(2, 115792086900472091959223405310199004229068084504756221892949117819703713273863L)
(3, 115792086900472091959223405310199004229068084504756221892949117819703713273863L)
(4, 115792086900472091959223405310199004229068084504756221892949117819703713273863L)
(5, 340282363487254643170864374573732807431L)
(6, 115792086900472091959223405310199004229068084504756221892949117819703713273863L)
(7, 340282363487254643170864374573732807431L)
(8, 115792086900472091959223405310199004229068084504756221892949117819703713273863L)


Voila. $p$ is 340282363487254643170864374573732807431 and $q$ is $N/p$.