Problem 12 - Algebra A Combined Function

Question

Identify each number as prime or composite. See Example \(3 .\) 21

Step-by-Step Solution

Answer

21 is a composite number.

Step 1: Understand Prime and Composite Numbers

A *prime number* is a natural number greater than 1 that has no positive divisors other than 1 and itself. A *composite number* is a natural number greater than 1 that has more than two positive divisors. That is, it can be divided exactly by numbers other than 1 and itself.

Step 2: List Divisors of 21

Let's find the divisors of 21. We know it ends with "1," and it is an odd number, so it is not divisible by 2. Check division by 3: a number is divisible by 3 if the sum of its digits is divisible by 3.

Step 3: Check Divisibility by 3

The sum of the digits of 21 is 2 + 1 = 3. Since 3 is divisible by itself, that indicates 21 is divisible by 3.

Step 4: Verify 21's Divisibility by Another Number

21 divided by 3 equals 7 (21 ÷ 3 = 7). Since 7 is a divisor and is different from 1 and 21, it confirms that 21 has divisors other than 1 and itself.

Step 5: Conclusion

Since 21 has divisors other than 1 and itself (specifically, 3 and 7), it is not a prime number. Therefore, 21 is a composite number.

Key Concepts

Number Divisibility Rules

Number divisibility rules are the guidelines that help us to quickly determine whether one number divides another without a remainder. These rules make the process of testing divisibility simpler and quicker than performing full division. To check if a number is divisible by a smaller number, we use these simple observations:

• A number is divisible by 2 if it is even, meaning the last digit is 0, 2, 4, 6, or 8.
• A number is divisible by 3 if the sum of its digits is divisible by 3. For example, for 21, add 2 and 1 to get 3, which is divisible by 3.
• A number is divisible by 5 if it ends in 0 or 5.
• A number is divisible by 10 if it ends in 0.

Utilizing divisibility rules can help to quickly assess if a bigger number can be divided evenly, without performing long division. This tool is particularly useful in identifying whether a number is a composite or prime.

Identifying Prime Numbers

Identifying prime numbers involves checking if they have exactly two distinct divisors: 1 and the number itself. Prime numbers are the building blocks of the number system. They are crucial in mathematics due to their fundamental role in number theory and various algorithms. For instance, the first few prime numbers are 2, 3, 5, 7, 11, and 13.

• 2 is the only even prime number since every other even number can be divided by 2.
• To identify if a number like 13 is prime, check divisibility by all prime numbers below its square root. If none can divide it fully, it is prime.

With practice and familiarity with small primes, identifying larger primes becomes less tedious. However, it's a task that requires careful checking of potential divisors.

Identifying Composite Numbers

Composite numbers are numbers that have more than two positive divisors. Pinpointing them involves finding any divisors other than 1 and the number itself. Using divisibility rules in step-by-step checks can help confirm if a number is composite.

• Numbers like 4, 6, 8, 9, and 10 are easy examples of composite numbers.
• For 21, since 2 is not a divisor (21 is odd), we move to the next possible number. It is divisible by 3 (since the sum 2 + 1 equals 3, divisible by 3) and 7 (since 21 divides exactly by 7), proving it composite.

Recognizing composite numbers can assist in a deeper understanding of numbers and their properties, as they can be broken down into smaller prime factors, aiding in tasks such as finding the least common multiple or greatest common divisor of sets of numbers.