Facts:
Any product of odd numbers is odd.
The sum of two odd numbers is even.
  
311 and 513 are both the product of odd numbers, 
so they are both odd

So 311 + 513 is the sum of two odd numbers and 
is therefore even.

Any even number is divisible by 2. Therefore 2 is the smallest prime
that divides 311 + 513.  [In fact, 2 is the
smallest prime, period!] 

[In fact 311 + 513 = 1220880272 = 24*47*107*15173, but we don't need that!)

Edwin