Problem 81 - Prealgebra

Question

Factor into prime factors. $$60$$

Step-by-Step Solution

Answer

The prime factorization of 60 is \(2^2 \times 3 \times 5\).

Step 1: Understanding Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime numbers. Prime numbers are positive integers greater than 1 that have no divisors other than 1 and themselves.

Step 2: Start Factorization

Begin by finding the smallest prime number that divides 60. The smallest prime number is 2. Check if 60 is divisible by 2.

Step 3: Divide by 2

Since 60 is an even number, it is divisible by 2. Perform the division: \[ 60 \div 2 = 30 \]Record the factor 2.

Step 4: Continue with 30

Repeat the process with 30, the quotient from the previous step. Check divisibility by 2 again, since it is also an even number.

Step 5: Divide by 2 Again

30 is divisible by 2. Perform the division: \[ 30 \div 2 = 15 \]Record the factor 2 again.

Step 6: Move to Next Prime

Now, take 15. Check divisibility by the next smallest prime number, which is 3.

Step 7: Divide by 3

15 is divisible by 3. Perform the division: \[ 15 \div 3 = 5 \]Record the factor 3.

Step 8: Check Remaining Number

The quotient is 5, which is a prime number itself. Therefore, the factorization process ends here. Record the factor 5.

Step 9: Final Product of Factors

Combine all recorded prime factors to write the complete prime factorization of the original number: \[ 60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5 \]

Key Concepts

Prime Numbers

Prime numbers are the building blocks of all numbers. They are unique because they can only be divided by 1 and themselves without leaving a remainder. For instance, the number 2 is a prime because the only whole numbers that multiply to produce 2 are 1 and 2. It’s important to be familiar with the list of prime numbers to make factorization easier. Common small primes include:

• 2
• 3
• 5
• 7
• 11
• 13

Understanding this concept is crucial for breaking down composite numbers in the factorization process. Remember, any whole number greater than 1 is either a prime or can be broken down into a product of primes.

Divisibility

Divisibility refers to whether a number can be divided by another number without leaving a remainder. This concept is essential in the prime factorization process. To test divisibility:

• By 2: The number must be even.
• By 3: Sum of the digits of the number must be divisible by 3.
• By 5: The number must end in 0 or 5.

In our example with the number 60, we started with divisibility by 2 (since it’s even), divided by 2 multiple times, then checked divisibility by 3, and finally ended up with the prime number 5.

Product of Primes

When we express a number as a product of primes, we are breaking it into its simplest components. For example, the number 60 can be broken down into:

• 60 = 2 × 2 × 3 × 5

This means that instead of thinking of 60 as just a whole number, we understand it as a product of individual prime numbers. In exponential form, 60 is expressed as:

• 60 = 2^2 × 3 × 5

This representation is helpful not only in simplifying calculation processes but also in recognizing patterns in numbers, making them easier to work with in various mathematical contexts.

Factorization Process

Factorization process involves breaking down a composite number into prime numbers. It starts by selecting the smallest prime that can divide the number completely and continues until only prime numbers are left. Here’s how the process works practically:

• Step 1: Start with the smallest prime number. For even numbers, this is typically 2.
• Step 2: Divide the number by this prime and record the factor.
• Step 3: Repeat the process with the resulting quotient.

Repeat these steps until the quotient is a prime number or 1. In the case of 60, we divided by 2 twice, followed by dividing by 3, and ended with the prime number 5. The most important thing to remember is that the process ends once you obtain all prime factors, thus breaking the number into simpler parts.