Problem 13 - Algebra for College Students

Question

Factor each of the composite numbers into the product of prime numbers. For example, \(30=2 \cdot 3 \cdot 5 .\) $$44$$

Step-by-Step Solution

Answer

The prime factorization of 44 is \(2^2 \times 11\).

Step 1: Determine Smaller Prime Factor

To factor the composite number 44, start by identifying the smallest prime number that divides it. The smallest prime number less than 44 is 2. Check if 44 is divisible by 2: 44 divided by 2 equals 22. Thus, 2 is a factor of 44, and we can write 44 as a product: \(44 = 2 \times 22\).

Step 2: Factor Remaining Composite Number

Now focus on factoring 22. Again, start with the smallest prime number, 2. Divide 22 by 2, which equals 11. Thus, 2 is a factor of 22, and we can write 22 as a product: \(22 = 2 \times 11\).

Step 3: Confirm Prime Components

Verify that the remaining factor, 11, is a prime number. Since 11 has no divisors other than 1 and itself, it is a prime number. Therefore, the complete factorization of 44 into its prime components is confirmed as \(44 = 2 \times 2 \times 11\).

Key Concepts

Composite Numbers

Composite numbers are numbers that have more than two distinct positive divisors. In simpler terms, these are numbers that can be divided evenly by 1, themselves, and at least one other number. This contrasts with prime numbers, which have only two divisors: 1 and themselves.
Understanding composite numbers is crucial in factorization, as these numbers can be broken down into smaller parts. When we talk about factorization, especially prime factorization, we typically start with composite numbers.

• For instance, the number 44 is composite because it can be divided by 1, 44, 2, 4, 11, and 22.
• In this case, the objective is to rewrite 44 as a product of prime numbers, which leads us to the topic of the factorization process.

Prime Numbers

Prime numbers are like the building blocks of integers. They can only be divided by 1 and themselves without leaving any remainder. For example, numbers like 2, 3, 5, 7, and 11 are all prime.
Prime numbers play a vital role in mathematics due to their indivisibility by other numbers, which is fundamental in different mathematical areas, particularly in number theory and cryptography.

• The number 11 is prime because no other numbers divide it completely except for 1 and 11.
• Similarly, 2 is another example of a prime number, and importantly, it is the only even prime number.

Understanding prime numbers is essential because when you factor composite numbers, you'll often need to recognize these prime numbers in the factorization process.

Factorization Process

The factorization process involves breaking down a composite number into its prime components. It is a systematic process and is crucial in various mathematical computations.
Here’s a simple breakdown of how to handle the factorization process:

• Identify the Composite Number: Begin with the composite number you want to factor, like 44 in our example.
• Find the Smallest Prime Factor: Start by checking divisibility by the smallest prime number, which is usually 2. For 44, dividing by 2 gives 22.
• Repeat the Process: Next, take the resulting number and repeat the process. Divide 22 by 2 to get 11.
• Check for Primes: Finally, ensure each remaining factor is a prime number. In this case, 11 is prime.

Once complete, the prime factorization of 44 is \[44 = 2 \times 2 \times 11.\]This showcases the breakdown of a composite number into its fundamental, indivisible components.