A composite number is any number that isn't prime.

 The remaining numbers are prime.

Finding just one factor eliminates a number from the set of prime numbers. For a large number , begin with the smallest prime and exhaust all possible prime factors , , , , , etc. 

 with  remaining

 with  remaining

 with  remaining

 with  remaining

 with  remaining

 with  remaining

For bonus points, explain why it is only necessary to test prime factors up to and including .

 and  so .

 with  remaining.

For any number  ,  with  remaining .

Thus, for a number  to be a factor of , it must pair with another factor .

However, we have already exhausted all of the prime numbers  so no prime factor exists.

 which is the only rational number listed. 

 is real but not rational, that is, it cannot be expressed in the form  where  is the set of all integers and .

 and  are complex numbers and therefore not real. A complex number is any number that contains the expression  or .

A prime number is a number with with no integer divisors, or factors, other than  and the number itself. The prime factors of  are  and . (For technical reasons,  is not considered prime and the number itself is not considered a factor.) The prime factors of  are  and . The prime factor of  is . , however, cannot be divided evenly by any integer other than  and  thus is prime.

A positive number is any number that is greater than zero.  is a negative decimal,  is a negative quotient,  is non-negative but it is not strictly to the right of zero on the number line. 

, or pi, is a mathematical symbol for the ratio of a circle's circumference  divided by twice its radius  or

and, though irrational, is the only number listed that is strictly greater than zero. Irrational numbers are defined more rigorously later in the problem set.

This is a bit of a trick question.  is an irrational number and the incorrect answers listed are all approximations of . However, every number listed can be expressed as a quotient of integers  except . The square root of two is irrational as is the square root of any prime number. 

A rational number is a number that can be expressed in the form  where both  and  are integers. 

 which is not only rational but an integer.

. Any square root of a prime number, the number 2 here, is irrational.

Finally, any square root of a negative number is complex and not real.

Remember that  and that any number that includes either  or  is complex and therefore not real. 

 which is real but negative.

The other numbers evaluate as

 ,

 , and 

.

Only  is a positive, real number. 

All of the numbers given are complicated, but only one is complex in the mathematical sense. A complex number is a number expressed in the form  where  and  are real numbers (the formal, mathematical expression for this is   ) and . 

 so  and  is complex.

While this problem can be answered by straight division, an easier method of doing so would be to apply the division tests for 3 and 4.

An integer is divisible by 3 if and only if the sum of its digits is divisible by 3. The digit sum of the number is 

The digit sum is not divisible by 3. It follows that 176,176,176,176 is not divisible by 3.

An integer is divisible by 4 if and only if the integer formed by its final two digits is divisible by 4. This integer is 76, and

.

The two-digit integer is divisible by 4. It follows that 176,176,176,176 is not divisible by 4.