Problem 38 - Basic College Mathematics

Question

Determine whether each number is prime, composite, or neither. $$ 2 $$

Step-by-Step Solution

Answer

The number 2 is prime.

Step 1: Understanding Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Step 2: Understanding Composite Numbers

A composite number is a natural number greater than 1 that has more than two positive divisors.

Step 3: Understanding Neither Prime Nor Composite

The number 1 is neither prime nor composite as it only has one positive divisor, which is itself.

Step 4: Analyzing the Number 2

The number 2 has only two positive divisors: 1 and 2.

Step 5: Conclusion

Since 2 has exactly two positive divisors, it fits the definition of a prime number.

Key Concepts

Prime Number Definition

A prime number is a natural number that is greater than 1 and has no divisors other than 1 and itself.
They are the building blocks of all numbers because every number can be factored into primes.
Here are some key points:

• A prime number has exactly two distinct positive divisors.
• Common examples include 2, 3, 5, 7, and 11.
• The number 2 is unique because it is the only even prime number.

Understanding these properties can help you identify primes quickly.
For example, look at the number 2:
- It has two divisors: 1 and 2.
- No other numbers divide it without leaving a remainder.
This makes 2 a prime number.

Composite Number Definition

A composite number is a natural number greater than 1 that has more than two positive divisors.
This means that composite numbers can be divided by numbers other than 1 and themselves.
Here's what you should remember:

• Composite numbers have at least one divisor other than 1 and itself.
• Common examples include 4, 6, 8, 9, and 12.
• They can always be factored into smaller integers.

Let's take the number 4 as an example:
- Its divisors are 1, 2, and 4.
- Unlike prime numbers, it can be divided by 2 as well.
This makes 4 a composite number.

Number Analysis

When analyzing whether a number is prime or composite, follow these steps:

1. Check if the number is greater than 1.
2. Count how many positive divisors the number has.
3. If it only has two divisors (1 and itself), it's prime.
4. If it has more than two divisors, it's composite.

Consider the number 10:
- It has divisors 1, 2, 5, and 10.
- Since it has more than two divisors, 10 is composite.
For prime numbers, take 5 as an example:
- Its only divisors are 1 and 5.
- This makes 5 a prime number.
Finally, remember that 1 is neither prime nor composite because it only has one divisor, which is itself.