Number Theory

We can use handy tricks to solve this problem. 

Let's focus on the last digit of the sum. This will come in handy. 

- 5 raised to any power will end in 5

7's last digit follows a pattern

$7^1  = 7 \\ 7^2 = 49 \\ 7^3  = 343\\ 7^4 = 2401$

- Since it repeats in groups of 4 7^13  ends in 7

$5^19 + 7^13 + 23$    will end in  $5 + 7 + 3 =  5$

This means 2 cannot be a divisor. 

The smallest  prime divisor wil either be 3  or 5

$[5^19 + 7^13 + 23] \pmod 3  = 2 \\ [5^19 + 7613 + 23 ] \pmod 5 = 0$

The smallest prime divisor is 5

Thanks! :)