Problem 1 - Algebra for College Students

Question

Classify each number as prime or composite. $$63$$

Step-by-Step Solution

Answer

63 is a composite number because it has divisors other than 1 and itself.

Step 1: Define 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 more than two distinct positive divisors.

Step 2: Find Divisors of 63

To determine if 63 is prime or composite, we need to check if it has divisors other than 1 and 63. Start by checking divisibility by smaller prime numbers like 2, 3, 5, and 7.

Step 3: Check Divisibility by 2

63 is an odd number, so it is not divisible by 2.

Step 4: Check Divisibility by 3

To check if 63 is divisible by 3, sum its digits: 6 + 3 = 9. Since 9 is divisible by 3, 63 is divisible by 3. Thus, 63 ÷ 3 = 21.

Step 5: Verify More Divisibility

Since we found another divisor, 3, besides 1 and 63, this means 63 has more divisors. To confirm it's composite, we can check if 21 is composite as well, though it’s not strictly necessary to confirm 63 is composite.

Step 6: Conclude Classification

Since 63 has divisors other than 1 and itself (specifically, it is divisible by 3 and 21), 63 is a composite number.

Key Concepts

Prime Numbers

Prime numbers are special because they have a unique quality about their divisors. A prime number is a whole number greater than 1. Importantly, it can only be divided evenly by 1 and itself.

This means that numbers like 2, 3, 5, 7, and 11 are all prime numbers. Each of these numbers doesn't break easily by other numbers apart from 1 and the number itself.

If a number has exactly two distinct positive divisors, it stands proudly as a prime number. But remember, the number must be greater than 1. This eliminates 1 from being a prime number.

• 2 is the smallest and the only even prime number
• Prime numbers are fundamental for math because they are the building blocks of the numbers
• Any number can be expressed as a product of prime numbers, a process called prime factorization.

Composite Numbers

Composite numbers are the ones that don't fit the prime profile because they have more than two distinct positive divisors. Unlike prime numbers, composite numbers can be divided by numbers other than just 1 and itself.

For example, let's look at 4. It can be divided not only by 1 and 4 but also by 2. That makes 4 a composite number. The same goes for numbers like 6, 9, 15, and 21.

What's interesting is that the concept of composites essentially groups all the non-prime numbers (except for 1) together. When you find more than two ways to divide a number evenly, you're dealing with a composite number.

• Composite numbers must have more than two factors
• They include at least one even number other than 2
• Every composite number can be factored into smaller integers

Divisibility Rules

Divisibility rules are nifty shortcuts that help you determine if one number can be divided by another without leaving a remainder. These rules make math calculations much faster, especially when working with larger numbers.

Let's take the rule for 3: If the sum of a number's digits is divisible by 3, then the number itself is divisible by 3. In our example of 63: The digits sum up to 9 (6 + 3), and because 9 is divisible by 3, so is 63.

Another useful rule is for 2: If a number is even, it's divisible by 2. However, since 63 is odd, it's not divisible by 2.

• For 5: A number is divisible by 5 if its last digit is 0 or 5
• For 10: A number is divisible by 10 only if it ends in 0
• Knowing divisibility rules speeds up factorization and helps identify prime and composite numbers quickly