Number Theory

The smallest possible prime divisor a number can have is of course, \(2\)

 

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. 

 

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