Problem 70 - Fundamentals of Mathematics

Question

Find the prime factorization of each of the whole numbers. 480

Step-by-Step Solution

Answer

The prime factorization of 480 is \( 2^5 \times 3 \times 5 \).

Step 1: Understand the Task

To find the prime factorization of 480, we need to express it as a product of prime numbers.

Step 2: Start with the Smallest Prime Number

The smallest prime number is 2. Divide 480 by 2. Since 480 is even, divide it by 2: 480 ÷ 2 = 240.

Step 3: Continue Dividing by 2

Divide 240 by 2 again, as it is still even: 240 ÷ 2 = 120.

Step 4: Divide by 2 Again

Continue this process for 120, as it is still even: 120 ÷ 2 = 60.

Step 5: Keep Dividing by 2

Divide 60 by 2 once more, as it remains even: 60 ÷ 2 = 30.

Step 6: Final Division by 2

Divide 30 by 2 again: 30 ÷ 2 = 15.

Step 7: Switch to the Next Prime, 3

Now that 15 is not divisible by 2, we move to the next smallest prime number, which is 3. Divide 15 by 3: 15 ÷ 3 = 5.

Step 8: Check for Remaining Prime Number

15 divided by 3 gives us 5, which is a prime number. We stop here as all factors are primes.

Step 9: Compile the Prime Factorization

Array the factors from all steps: 2 × 2 × 2 × 2 × 2 × 3 × 5.

Step 10: Write Using Exponents

Write the factors using exponents to simplify the expression:The prime factorization of 480 is: \( 2^5 \times 3^1 \times 5^1 \).

Key Concepts

Prime Numbers

Prime numbers are the building blocks of the number system. They are numbers greater than 1 that have no divisors other than 1 and themselves. This means a prime number cannot be formed by multiplying two smaller numbers. For example, 2, 3, 5, 7, 11, and 13 are all prime numbers. These numbers have only two factors: 1 and the number itself. Understanding prime numbers is crucial for prime factorization because each number is broken down into a product of primes.

• 2 is the smallest and the only even prime number.
• All other even numbers can be divided by 2, which makes them composite.
• Primes are infinite, meaning there's no largest prime number.

Prime numbers help us simplify math problems, especially when factoring large numbers into smaller components.

Exponents

Exponents are a way to express repeated multiplication of the same number. For instance, writing a number like 2 five times in multiplication form as 2 × 2 × 2 × 2 × 2 can be simplified using exponents: it becomes 2 raised to the power of 5 or \( 2^5 \). Exponents are helpful when working with prime factorization to represent factors concisely. This makes calculations and comparisons faster and error-free.

• \( a^n \) means a is multiplied by itself n times.
• The exponent "n" tells you how many times to multiply the base "a".
• Zero exponent rule: any non-zero number raised to 0 is 1 \( a^0 = 1 \).

Understanding exponents is important for expressing the power of prime factors in a compact form.

Factorization

Factorization is the process of breaking down a number into a product of other numbers or factors, which when multiplied together give the original number. Prime factorization specifically involves dividing a number into its prime components. This is useful for simplifying fractions, finding greatest common divisors, and solving many mathematical problems.
Here's how to factorize the number 480 into its prime factors:

• Start with the smallest prime, divide by 2 repeatedly because 480 is even, until you can't anymore.
• After dividing by 2, move to the next smallest prime, which is 3; continue dividing until you can't.
• Finally, you are left with prime numbers or 1, indicating completion.

The process results in expressing 480 as \( 2^5 \times 3^1 \times 5^1 \), which streamlines calculations in various mathematical applications.