Problem 74 - Prealgebra

Question

Determine whether each number is prime or composite. $$21$$

Step-by-Step Solution

Answer

21 is a composite number.

Step 1: Understanding Prime and Composite Numbers

A prime number is a number greater than 1 that has no divisors other than 1 and itself. A composite number is a number that has additional divisors besides 1 and itself.

Step 2: Checking Divisibility

To determine if 21 is composite, check if it has any divisors other than 1 and 21. 1. Check divisibility by 2: 21 is not even, so it's not divisible by 2. 2. Check divisibility by 3: Find the sum of the digits (2 + 1 = 3). Since 3 is divisible by 3, 21 is divisible by 3. 3. Since 21 is divisible by 3, it is not a prime number.

Step 3: Conclusion

Since 21 is divisible by a number other than 1 and itself (specifically 3), it is a composite number.

Key Concepts

Divisibility Rules

Divisibility rules are simple shortcuts to determine whether a number can be divided by another without leaving a remainder. These can save a lot of time when working with large numbers. For example, a number is divisible by:

• 2 if it is even, which means its last digit is either 0, 2, 4, 6, or 8.
• 3 if the sum of its digits is divisible by 3. For example, with the number 21, the sum of its digits is 2 + 1 = 3, and since 3 is divisible by 3, so is 21.
• 5 if it ends in 0 or 5.
• 10 if it ends in 0.

There are rules for other numbers as well, such as 4, 6, 7, 8, 9, and so on. Knowing these can help you quickly identify if a number is prime or composite by checking divisibility.

Prime Numbers

Prime numbers are the building blocks of the number system. A prime number is any whole number greater than 1 that has no divisors other than 1 and itself. This means it can't be evenly divided by any other numbers. Examples of prime numbers are 2, 3, 5, 7, 11, and 13, among others.

What makes prime numbers special is that every whole number greater than 1 is either a prime number or can be factored into prime numbers. For instance, 6 can be broken down into the factors of 2 and 3, which are both prime. Using divisibility rules can help identify if a number is prime. If it can't be divided evenly by any of the prime numbers smaller than itself, it's likely prime.

Composite Numbers

Composite numbers are all about having more than two factors. Unlike prime numbers, composite numbers can be divided evenly by numbers other than 1 and itself. This means they have additional divisors or factors.

For instance, the number 21, which was the focus of the original exercise, is a composite number. To confirm, we checked its divisibility:

• It is divisible by 3 because the sum of 21's digits equals 3, which is divisible by 3.

Thus, 21 has divisors other than just 1 and itself, namely 3 and 7, because 21 equals 3 x 7. Knowing how to identify composite numbers can be useful in many areas, from breaking down fractions to simplifying equations.