Problem 16 - Algebra for College Students

Question

Factor each of the composite numbers into the product of prime numbers. For example, \(30=2 \cdot 3 \cdot 5 .\) $$64$$

Step-by-Step Solution

Answer

64 is factored into prime numbers as \(2^6\).

Step 1: Recognize 64 is a Composite Number

Since 64 is more than 1 and not a prime number, we know it can be divided into simpler components that are prime numbers.

Step 2: Start with the Smallest Prime Number

Begin by dividing 64 by the smallest prime number, which is 2. Check if 64 is divisible by 2. Since 64 is even, it is divisible by 2.

Step 3: Continue Dividing by 2

Divide 64 by 2: \[ \frac{64}{2} = 32 \]32 is also divisible by 2, so we continue this process.

Step 4: Repeat Division

Keep dividing by 2 until you reach a quotient that is not divisible by 2: \[ \frac{32}{2} = 16 \]\[ \frac{16}{2} = 8 \]\[ \frac{8}{2} = 4 \]\[ \frac{4}{2} = 2 \]\[ \frac{2}{2} = 1 \]

Step 5: Compile the Prime Factors

Since we began with 64 and ended at 1, the division steps involved only the prime number 2. We divided by 2 six times, so the factorization can be written as:\[ 64 = 2^6 \]

Key Concepts

Composite Numbers

Composite numbers are fascinating and widely discussed when learning about prime factorization. Unlike prime numbers, which have only two distinct divisors (1 and themselves), composite numbers have more than two factors. For example, the number 64 is a composite number.

To determine if a number is composite:

• The number must be greater than 1.
• It should have divisors other than 1 and itself.

Recognizing a composite number is the first step in the division process, which will help break it down into its prime factors, as illustrated in many math problems.

Prime Numbers

Prime numbers play a crucial role in the prime factorization process. They are the natural building blocks for all numbers since they cannot be divided by any other numbers except 1 and themselves without yielding a remainder. In the problem we're considering, prime numbers are used as tools to simplify the original composite number, 64. It is important to remember that a prime number's simplicity is its strength.

When engaging in factorization:

• Start with the smallest prime number, which is 2.
• Continue with other primes (3, 5, 7, etc.) if needed.

In the case of our example, repeatedly dividing by the smallest prime number was all that's needed to find the prime factors of 64, resulting in six instances of the number 2.

Division Process

The division process is essential to breaking down a composite number into its prime factors. This method is both systematic and straightforward, ensuring that each step logically leads to the factorization of the composite number.

Here's how to apply the division process effectively:

• Start by dividing the composite number by the smallest prime number that fits evenly (no remainder).
• Continue dividing the quotient by the same prime, as long as it is divisible.
• Once it's not divisible by that prime, move to the next smallest prime.

For example, in factoring 64, we began with 64 divided by 2 to get 32. Since 32 is still even, we continued dividing by 2 until reaching 1, showing that 64 equals \(2^6\). This division process can be effortlessly followed and reproduced for any composite number, leading to an easy and accurate factorization.