Problem 69 - Fundamentals of Mathematics

Question

Find the prime factorization of each of the whole numbers. 176

Step-by-Step Solution

Answer

The prime factorization of 176 is \( 2^4 \times 11 \).

Step 1: Start with the Smallest Prime Number

Begin the factorization process by dividing the number 176 by the smallest prime number, which is 2. Since 176 is even, it can be divided by 2. Doing so gives: \[ 176 \div 2 = 88 \].

Step 2: Continue Dividing by 2

Check if 88 can be divided by 2. It is also an even number, so divide again by 2: \[ 88 \div 2 = 44 \].

Step 3: Divide by 2 Again

Continue dividing by 2 since 44 is even. Perform the division: \[ 44 \div 2 = 22 \].

Step 4: Final Division by 2

Divide 22 by 2 as it is still even:\[ 22 \div 2 = 11 \].

Step 5: Identify Remaining Prime Number

Now, check if 11 is a prime number. Since 11 is only divisible by 1 and itself, 11 is a prime number.

Step 6: Compile the Prime Factors

Summarize the factors obtained from the divisions. The complete prime factorization of 176 is \[ 2^4 \times 11 \].

Key Concepts

Whole Numbers

Whole numbers are the set of numbers that include all positive numbers, including zero, without any fractions or decimals. These numbers are essential in math as they are the building blocks of more complex operations.
Whole numbers include numbers like:

• 0
• 1
• 2
• 3, and so on.

Whole numbers do not include negative numbers, decimal numbers, or fractions.
When performing operations such as addition, subtraction, or multiplication, the results often remain whole numbers. However, when dividing whole numbers, like in our prime factorization exercise involving the number 176, it's possible that the result is not a whole number, leading to the need for division through factors like prime numbers.

Prime Numbers

Prime numbers are a fascinating area of mathematics. They are numbers greater than 1 that have no divisors other than 1 and themselves. This means they cannot be divided evenly by any other numbers.
Some of the smallest prime numbers include:

• 2 (the only even prime number)
• 3
• 5
• 7, and so on.

Prime numbers play a crucial role in various fields of mathematics, including number theory and cryptography. They are also fundamental in finding the prime factorization of whole numbers, like the number 176. By systematically dividing by prime numbers, we can break a whole number down into its building blocks. Understanding this concept helps simplify complex problems and equations.

Division Process

The division process is a method used to break down numbers into simpler components. Specifically, when determining the prime factorization of a number, division is key.
To use division for prime factorization:

• Start by identifying the smallest prime number that can divide the given number.
• In our exercise, we begin with 2 as it's the smallest prime number and 176 is even.
• After each division, check if the resulting quotient can be divided by the same or a different prime number.

Repeat this process until no further division is possible, ending with a prime number. Through division, we ensure that each factor is prime, crucial for the complete prime factorization. This method effectively helps in simplifying and solving mathematical problems.

Mathematical Steps

Mathematical steps are the sequential procedures or actions taken to solve a problem. When tackling exercises like prime factorization, it's important to follow each step meticulously to arrive at the correct solution.
Here’s a simplified breakdown of the steps involved in the prime factorization of 176:

• Step 1: Begin by dividing 176 by the smallest prime, 2, resulting in 88.
• Step 2: Divide 88 by 2, yielding 44.
• Step 3: Again, divide 44 by 2 to get 22.
• Step 4: Divide 22 by 2 to achieve 11, a prime number.
• Step 5: Recognize 11 as a prime number, completing the factorization.

The final outcome is the prime factorization: 2 raised to the power of 4 multiplied by 11, expressed as \( 2^4 \times 11 \).
These steps are straightforward, allowing a structured path to the solution, which aids in understanding and learning mathematical concepts.