Number Theory

We can solve this problem by looking at the last digit of the numbers. 

A power of 5 always ends with 5, so 5^19 ends with 5. 

We also know that  \(7^{13}=96889010407\) which ends with 7. 

So, we we add up the last digits, we get \(7+5+3 = 15\)

 

Since it ends with 5, the number must be divisble by 5. 

We also know that it can't be divisble by 2 and 3 since neither of \(5^{19} , 7^{13} , 23\) are divisble. 

 

So our final answer is 5. 

 

Thanks! :)