Problem 11 - Elementary Algebra

Question

Determine the GCF of all the terms. $$ 15 x, 30 $$

Step-by-Step Solution

Answer

The GCF is 15.

Step 1: Identify the Factors

Break down each given term into its prime factors. For 15x, the factors are 3, 5, and x. For 30, the factors are 2, 3, and 5.

Step 2: Find Common Factors

List the factors common to each term. The common factors between 15x (3, 5, x) and 30 (2, 3, 5) are 3 and 5.

Step 3: Calculate the GCF

Multiply the common factors. The GCF is calculated as follows: 3 * 5 = 15.

Key Concepts

Prime Factorization

Prime factorization is a method used to break down a whole number into its basic building blocks, which are prime numbers. A prime number is any number greater than 1 that has no positive divisors other than 1 and itself. To perform prime factorization, you simply keep dividing the number by the smallest prime number until you're left with 1.

In the exercise, for the term 15x, we ignore the variable (x) and focus on the number 15. We find the prime factors of 15 by dividing it: 15 divided by 3 equals 5, and both 3 and 5 are prime numbers. So, the prime factorization of 15 is 3 and 5.

For the number 30, the process is similar. Start by dividing by the smallest prime number that fits, 2. Since 30 divided by 2 equals 15, and we already know the prime factors of 15, we have our prime factors of 30 as 2, 3, and 5.

Common Factors

Common factors are the numbers that appear in the prime factorization of both numbers. To find these, list all the factors of each number, then identify the numbers that appear in both lists.

In the given problem, we found the prime factors of 15 and 30.

• 15: 3, 5
• 30: 2, 3, 5

The intersection of these lists reveals the common factors. Here, the numbers 3 and 5 are repeated in both sets. Identifying these common factors is crucial in determining the Greatest Common Factor (GCF).

Understanding common factors helps us simplify expressions or fractions by removing the greatest shared factor, making calculations more manageable.

Multiplication

Multiplication is the arithmetic operation of scaling one number by another. It's a fundamental concept used to combine numbers and find totals or sizes of groups. When calculating the GCF, multiplication is employed to combine the common factors we identified.

In the exercise, after identifying 3 and 5 as the common factors of both terms, we then multiply them together:

• 3 times 5 equals 15

Thus, we determine that the GCF of 15x and 30 is 15.

Understanding and using multiplication in this context allows us to efficiently calculate the greatest common factor by productively combining the shared factors of each term.