Number Theory

First, let's make a few observations. 

A power of 5 always ends with a 5. 

\(7^{13}=96889010407\), so 7^13 ends on a 7, 

And 23 ends with a 3. 

So we have \(7+5+3 = 15\). This means that \(5^{19} + 7^{13} + 23\) is divisible by 5.

Since 5, 7, and 23 are all odd, it's not possible the sum is divisble by 2. 

Also, since none of the terms are divisible by 3, the sum is not divisble by 3. 

Thus, 5 is our answer. 

Thanks! :)