The smallest prime number is actually .  is not a prime nor a composite number. It is a unit. 

A prime number is a number with factors of one and itself.

Let's try to find the factors.

It may not be easy to see  as a composite number, but if you know the divisibility rule for  which is double the last digit and subtract from the rest, you will see  is not prime. The divisibility rule for  is add the outside digits and if the sum matches the sum then it is divisible . The divisibility rule for  is if the digits have a sum divisible by , then it is . All even numbers are composite numbers with the exception of . So with these analyses, answer is . 

Since all of the digits don't add to a sum of , and we dont see any numbers een or end in , let's try the divisibility rule for  which is double the last digit and subtract from the rest.

Only  is divisible by  and is not prime and therefore our answer. 

The smallest prime number is actually .  is not a prime nor a composite number. It is a unit. This will eliminate the choices with a  in them. The next prime numbers are . Our answer is then .  is a perfect square and has more than two factors .

The order of the prime numbers start from .  is not prime as it's a unit.  is a composite number. So our fourth smallest prime number is .

Since all the numbers are odd and don't end with a , let's check the basic divisbility rule. The divisibility rule for  is if the digits have a sum divisible by , then it is. 

Based on this analysis, only  is divisible by  and therefore not prime and is our answer. 

Prime numbers are integers. So doing division and square roots will not generate integers. By doing multiplication and exponents, we involve more factors. The only possibility is subtraction. If  was  and we subtracted  we get  which is also prime. 

Other than the number itself and one, we also need to have prime factors in that number. Since it's not a perfect square, we need to find the smallest possible prime numbers. That will be  or  which is our answer. 

Since all the numbers are odd and don't end with a , let's check the basic divisbility rule. The divisibility rule for  is if the digits have a sum divisible by , then it is. 

Based on this analysis, only  is divisible by  and therefore not prime and is our answer. 

Let's work backwards. All even numbers are not prime so we skip .  is clearly divisible by .  is definitely a composite number as it's divisible by .  is divisible by  because of the divisibility rule . is divisbile by . The divisibility rule for   is double the last digit and subtract from the rest . From the remaining answers,  is prime and is the largest prime number under  and is our answer.