The prime numbers are 2,3,5,7,11,13,17,19,23,29,31,37,41,43,...

Get a calculator and start dividing 221 by these prime
numbers until you find one that doesn't have a decimal:

221÷2=110.5         That has a decimal, so we keep going:
221÷3=73.66666667   That has a decimal, so we keep going:
221÷5=44.2          That has a decimal, so we keep going:
221÷7=31.57142857   That has a decimal, so we keep going:
221÷11=20.09090909  That has a decimal, so we keep going:
221÷13=17           Aha!  No decimal! And the quotient, 17,
is a prime number itself!

So the answer is:  The two prime numbers you are looking
for are 13 and 17.  

Checking: Multiply them 13x17 = 221.  So their product is
221.  The only factors of 13 are 13 and 1.  The only factors
of 17 and 1.  So they have only one common factor, namely 1.
And since it is the ONLY common factor is has to be the 
GREATEST common factor.

So 13 and 17 are the two prime numbers you were looking for.

Edwin