Problem 6 - Algebra for College Students

Question

Classify each number as prime or composite. $$69$$

Step-by-Step Solution

Answer

69 is a composite number.

Step 1: Understand 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 greater than 1 that has additional divisors.

Step 2: Check for Divisibility by 2

Since 69 is not an even number, it is not divisible by 2.

Step 3: Check for Divisibility by 3

Sum the digits of 69: 6 + 9 = 15. 15 is divisible by 3, indicating that 69 is also divisible by 3. This means that 69 has additional divisors other than 1 and itself.

Step 4: Conclusion Based on Divisibility

Since 69 can be divided by 3 (other than 1 and 69), it is classified as a composite number.

Key Concepts

Divisibility Rules

Divisibility rules help us quickly determine if a number is divisible by another without performing long division. These rules are shortcuts to identify factors of a number. Here are some basic divisibility rules for small numbers:

• **Divisibility by 2**: A number is divisible by 2 if it is even, meaning it ends in 0, 2, 4, 6, or 8.
• **Divisibility by 3**: A number is divisible by 3 if the sum of its digits is divisible by 3. For example, for the number 69, sum the digits: 6 + 9 = 15. Since 15 is divisible by 3, so is 69.
• **Divisibility by 5**: A number is divisible by 5 if it ends in 0 or 5.
• **Divisibility by 10**: A number is divisible by 10 if it ends in 0.

Using these rules is like having a special tool to try out basic divisions mentally. They simplify the process of figuring out if a number has divisors other than 1 and itself, helping us classify it as prime or composite really quickly.

Prime Numbers

Prime numbers are the building blocks of mathematics. A prime number is one that can only be divided by 1 and itself without leaving a remainder. For example, numbers like 2, 3, 5, and 7 are prime numbers. They cannot be divided evenly by any number other than 1 or themselves.

Some characteristics of prime numbers include:

• **Greater than 1**: The definition explicitly excludes 1 from being a prime number.
• **No factors other than 1 and itself**: If a number has more divisors, it turns into a composite number.
• **The smallest prime number is 2**: Also, it's the only even prime number. All other even numbers can be divided by 2, which means they have at least one other divisor.

Understanding prime numbers is crucial because they are used to form all other numbers by multiplication. Any whole number can be decomposed into a set of prime factors, which makes the study of prime numbers fundamental in number theory.

Composite Numbers

Composite numbers are all about sharing. Unlike prime numbers, composite numbers have more than two divisors. This means composite numbers can be totally divided by numbers other than just 1 and themselves. Let’s take 69 as an example. By using divisibility rules, we found that when you add its digits (6 + 9), it sums up to 15. Since 15 is divisible by 3, this implies that 69 is also divisible by 3.

A few key points about composite numbers:

• **Greater than 1**: Like prime numbers, a composite number is always greater than 1.
• **Multiple divisors**: It must have divisors other than 1 and itself, such as 69, which has divisors 1, 3, 23, and 69.
• **Every even number greater than 2 is composite**: Because even numbers are divisible by 2, they automatically have a divisor in addition to 1 and itself.

Identifying whether a number is composite is important in understanding its properties and possible factorizations. This makes composite numbers a key area to explore in mathematics, especially in fields like algebra and arithmetic.