The key in public-key encryption is based on a hash value. This is a value that is computed from a base input number using a hashing algorithm. Essentially, the hash value is a summary of the original value. The important thing about a hash value is that it is nearly impossible to derive the original input number without knowing the data used to create the hash value. Here's a simple example:
Input number Hashing algorithm Hash value
10,667 Input # x 143 1,525,381
You can see how hard it would be to determine that the value 1,525,381 came from the multiplication of 10,667 and 143. But if you knew that the multiplier was 143, then it would be very easy to calculate the value 10,667. Public-key encryption is actually much more complex than this example, but that's the basic idea.
Public keys generally use complex algorithms and very large hash values for encrypting, including 40-bit or even 128-bit numbers. A 128-bit number has a possible 2128, or 3,402,823,669,209,384,634,633,746,074,300,000,000,000,000,000,000,000,000,000,000,000,000 different combinations -- this would be like trying to find one particular grain of sand in the Sahara Desert