Problem 13 - Elementary and Intermediate Algebra

Question

Label each of the following numbers as prime, composite, or neither. $$ 25 $$

Step-by-Step Solution

Answer

25 is a composite number.

Step 1: Define Prime, Composite, and Neither

A prime number has exactly two distinct positive divisors: 1 and itself. A composite number has more than two distinct positive divisors. If a number is neither prime nor composite, it usually means the number is 0 or 1.

Step 2: Check if the Number is Prime

Determine if 25 is a prime number by checking if it has any divisors other than 1 and itself. If it does, it is not prime.

Step 3: List Divisors of 25

Find the divisors of 25. We see that 25 is divisible by 1, 5, and 25, which means it has more than two divisors.

Step 4: Conclusion

Since 25 has divisors other than 1 and itself (specifically, 5), it is not a prime number. Since it has more than two divisors, it is a composite number.

Key Concepts

Prime Numbers

Prime numbers are very special in mathematics. They have exactly two distinct positive divisors: the number 1 and the number itself. This means they can't be divided evenly by any other number without leaving a remainder. For instance:

• **2**: The only divisors are 1 and 2, making 2 the smallest and only even prime number.
• **3**: It has no divisors other than 1 and 3.
• **5**: Divisible only by 1 and 5.

Prime numbers play a crucial role in number theory due to their fundamental properties in arithmetic. They are used in various fields such as cryptography, where they form the basis of encryption algorithms.

Composite Numbers

Unlike prime numbers, composite numbers have more than two distinct positive divisors. This means they can be divided evenly by numbers other than 1 and themselves. For example, the number 25:

• **Divisors of 25**: 1, 5, and 25.
• Since 25 has divisors other than just 1 and itself, it is considered a composite number.

Other examples of composite numbers are:

• **4**: Divisible by 1, 2, and 4.
• **6**: Divisible by 1, 2, 3, and 6.
• **9**: Divisible by 1, 3, and 9.

Understanding composite numbers helps us in various math problems, including factorization and finding common multiples.

Divisors

A divisor is a number that divides another number completely without leaving a remainder. For any number, divisors include 1 and the number itself, and potentially other numbers when dealing with composites. Here’s how to identify divisors:

• **List of Divisors**: For a given number, list all numbers that evenly divide into it.
• **Checking for Remainders**: If the division results in a remainder, it is not a divisor.

For example, let's look at 25 again:

• **Step 1**: Divide 25 by numbers starting from 1 up to 25.
• **Step 2**: Note down the numbers that evenly divide 25.
• **Result**: Divisors of 25 are 1, 5, and 25.

Recognizing divisors helps determine whether a number is prime, composite, or neither. This is fundamental in many areas of mathematics, such as solving equations and simplifying fractions.