Then define the public modulus m to be:
m = p × q
Even though m is public, no one knows how to factor a 200-digit number
in a reasonable amount of time.
So no one can figure out what p and q are.
Now use p and q to compute the secret signing exponent:
e = f(p,q)
See the notes at the end of chapter 4 if interested in the details of f.
If p and q have certain other properties,
then for any x < m,
x = (xe mod m)3 mod m
which is what we need for the signing and verification functions to be
inverses.
There are more details, but these are the essentials of the RSA Encryption Algorithm.