Problem 19 - Algebra for College Students

Question

Factor each of the composite numbers into the product of prime numbers. For example, \(30=2 \cdot 3 \cdot 5 .\) $$87$$

Step-by-Step Solution

Answer

87 is factored into primes as 3 and 29.

Step 1: Find the smallest prime number

Identify the smallest prime number that can divide 87. Start by testing 87 with the smallest primes: 2, 3, and so on. Since 87 is odd, it is not divisible by 2. The sum of the digits of 87 is 8+7=15, which is divisible by 3. Therefore, 87 is divisible by 3.

Step 2: Divide by the smallest prime

Divide 87 by 3 to find the other factor: \[87 \div 3 = 29\]This gives us the factors 3 and 29.

Step 3: Confirm the remaining factor is prime

Check if 29 is a prime number. A prime number is only divisible by 1 and itself. 29 is not divisible by any prime numbers less than its square root (which is approximately 5.39). Thus, 29 is a prime number.

Step 4: Write the complete factorization

Now that we've verified all factors are prime numbers, we write the complete prime factorization: \[87 = 3 \cdot 29\]

Key Concepts

Composite Numbers

Composite numbers are numbers that have more than two distinct positive divisors. This means they can be divided evenly by numbers other than just 1 and themselves. For example, 4, 6, and 8 are composite numbers because they can be divided by numbers such as 2 or 4 other than just 1 and themselves.

Here's a simple way to understand composite numbers:

• They are not prime numbers.
• They have factors other than 1 and the number itself.
• They start from the number 4, as it's the smallest composite number.

Recognizing composite numbers is crucial because factorizing them into their prime components helps us understand their structure better. This is often used in finding the greatest common divisor or simplifying fractions.

Prime Numbers

Prime numbers are fascinating and essential in mathematics. They can only be divided by 1 and themselves, which makes them the building blocks of other numbers.

To identify prime numbers, consider the following characteristics:

• A prime number has exactly two distinct positive divisors: 1 and itself.
• It cannot be made by multiplying two smaller positive integers.
• The number 2 is the smallest and the only even prime number.
• Numbers like 3, 5, 7, 11, and 13 are classic examples of primes.

These properties make prime numbers unique. No composite number can exist without prime factors, and identifying these factors, like finding 3 and 29 are factors of 87, is a skill you'll use often in mathematics.

Divisibility Rules

Divisibility rules make it easier to determine whether a number can be divided evenly by another without doing long division. These rules are like shortcuts. They help you quickly identify factors of numbers and decide if a division will result in an integer.

Some key divisibility rules include:

• Divisibility by 2: A number is divisible by 2 if it is even (i.e., 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 instance, 87 is divisible by 3 because 8 + 7 equals 15, and 15 is divisible by 3.
• 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.

These rules save time and are especially useful when you're factoring numbers or simplifying complex equations. Understanding divisibility helps find prime factors, as was done with 87 by initially checking divisibility by 3.