Prime And Composite Numbers

A composite number is a number that can be factored into two or more prime numbers. However, this factorization is always unique, as per the Fundamental Theorem of Arithmetic, which states that every composite number can be written as a product of prime numbers in a unique way, except for the order of the prime factors. For example:


In each case, the product of prime factors is unique (though the order of multiplication can vary). Thus, a composite number can always be written as a product of prime numbers in a unique way.

The correct answer is option (A): a unique way

Correct answer: Set of months in a year

Explanation:
A finite set is a set that has a countable number of elements.

- The set of all natural numbers is infinite, as natural numbers go on without end.
- The set of all prime numbers is also infinite, since there is no largest prime number.
- The set of months in a year has exactly 12 elements, so it is finite.

Therefore, the set of months in a year is the only finite set among the given options.

Correct answer: {2}

Explanation:
A prime number is a number that has only two distinct positive divisors: 1 and itself.

An even number is a number that is divisible by 2.

2 is the only number that is both even and prime because:
- It is divisible only by 1 and 2
- It is even

Hence, the only even prime number is 2, and the set is {2}.

Correct answer: 9

Explanation:
A composite number has more than two factors (not a prime number).

Let's check each option:
- 3 → Prime number
- 5 → Prime number
- 7 → Prime number
- 9 → Factors: 1, 3, 9 → more than two factors

So, 9 is the smallest odd composite number.