Just a Prime

Since we want to find the smallest prime dividing the sum, we start with the smallest prime number and move up. Many of them make mistake that 1 would be be answer because 1 is not a prime number. S first, we try 2.

Notice that $3^{11}$ and $5^{13}$ are both odd, so their sum must be even. This means that 2 must divide $3^{11}+5^{13}$ , and so since 2 is the smallest prime, our answer must be 2.

Since product of odd numbers gives odd number and their sum gives even number, which is divisible by smallest only even prime number that is 2.