Number Theory

The smallest possible prime divisor a number can have is of course, 

If we check the multiplication, we can see that 3 odd numbers can't possibly mulitply to be an even. 

So, $5^{19} * 7^{13} * 3^{31}$ is not divisble by 2. 

The next smallest is 3. 

The third term of the multiplication is $3^{31}$

This means, no matter what, this number has to be divisible by 3. 

So 3 is our final answer. 

Thanks! :)