Building a public-key and private-key:

Select two prime numbers:
>>> p = 97
>>> q = 233

Compute n and r:
>>> n = p * q
>>> n
22601
>>> r = (p-1)*(q-1)
>>> r
22272

Find e such that e and r are relatively prime (share no common factors):
>>> e = 2
>>> while (r % e == 0):
...     e = e + 1
... 
>>> e
5

Find d such that (e*d)%r equals 1:
>>> d = 1
>>> while (e * d) % r != 1:
...     d = d + 1
... 
>>> d
8909

Our public-key is (e,n): (5, 22601)
Our private-key is (d,n): (8909, 22601)

Someone encodes message M for us using our public key:
>>> M = 15110
>>> M**e % n
3082
(Message is 15110, they send 3082 to us)

We decode the coded message C using our private key:
>>> C = 3082
>>> C**d % n
15110

We don't tell anyone our private key so if someone
intercepts the transmitted message (3082), they won't
know how to decode it without knowing d.

To discover d, one would have to determine how to
factor n into its two primes. When p and q are
hundreds of digits long, this process of factoring
n is computationally hard to do which makes this
hard to crack.