A prime number is a number that havs factors of one and itself. Let's eliminate some choices.

A number divisible by  will have a ones digit of  or . So we eliminate . 

Even numbers are having ones digit end in . So we eliminate 

For the remaining , let's try the divisibility rule of . The sum of the digits add up to a multiply of  is diisible by . 

 This is good. 

 This is divisible by . 

 This fails as this is our answer. 

Positive integers start from one onward. No matter what number you multiply with, the answer is always positive and an integer. A positive integer multiplied by a positive integer is always a positive integer. 

For a number to be rational, the numerator and denominator have to be integers. The exception is the denominator must not equal zero. It can be negative, positive, or zero. 

Although all the choices have radicals and we know many radicals are irrational numbers, we can definitely simplify one of them.  is a perfect sqaure. The answer to that radical is .  is an integer and is our answer. 

It is true when we divide two rational numbers, we should get a rational number. The exception is when let's say we divide  and . If the denominator is , the answer is undefined. Undefined is a class of its own. This is the only exception so our answer is mostly rational numbers. 

A number is rational if it is possible to express it as a fraction with integers as the numerator and denominator. 25 and -1 are integers, and could be written as fractions: . is already expressed as a fraction. At first glance,

 looks like an irrational number becasue it repeats forever. This is different from an irrational number because an irrational number will go on forever but without a repeating pattern.  could actually be expressed as the fraction . This leaves us with the correct answer of which goes on forever [is non-terminating] but without any clear pattern or repeating part.

Let's list all the counting numbers which start from . 

 is neither prime nor composite. It is a unit; therefore, this answer isn't right. 

 are prime numbers.  A prime number only has factors of one and itself. Thus, these choices are not the correct answer.

 is a composite number. It has factors of . This is the smallest composite number and is our answer. 

 is not the correct answer because it is a composite number with factors of  and it is greater than the composite number . Thus, it cannot be the smallest composite number of the series.

Although all of the answers are radicals, many of them can be simplified as they are perfect squares. 

 is simplified to .

 is simplified to . 

 is simplified to .

 is simplified to .

Only  is not a perfect square and is irrational. Therefore, it is the correct answer.

Cubic number is a number raised to a third power. We should begin by listing the first few cubic numbers. The first few cubic numbers are  or . In doing so, we find that there is a match in our answer set which is . 

Composite numbers have at least one factor other than one and itself. For example,  is a composite number. The factors are .