Problem 53 - Algebra 1: Concepts and Skills

Question

Find the greatest common factor. $$ 49,64 $$

Step-by-Step Solution

Answer

The greatest common factor of 49 and 64 is 1.

Step 1: Factor each number

Start by factoring each number into its prime factors. The prime factors of 49 are 7 and 7 while the prime factors of 64 are 2, 2, 2, 2, 2, 2.

Step 2: Compare the factors and find the common one

Now we compare the prime factors of 49 and 64 to find the greatest factor that they share. Unfortunately, in this case, they do not share any prime factors.

Step 3: State the greatest common factor

Since there are no common prime factors between 49 and 64, the greatest common factor (GCF) is 1. Whenever two numbers do not share any prime factors, their GCF is always 1.

Key Concepts

Factoring Numbers

Factoring numbers is a foundational skill in mathematics that involves breaking down a number into a product of other numbers, preferably primes. Think of it like deconstructing a building into its individual bricks. When we factor numbers, we are seeking to simplify the number into its basic building blocks.

For instance, when factoring the number 12, we can break it down into 2 × 6 or further into 2 × 2 × 3. These smaller numbers are the factors of 12. The factoring process can be simple for small numbers, but as numbers grow larger, it might become more complex. To approach factoring efficiently, one strategy is to start with the smallest prime number that divides the number and continue the process until we end up with all prime factors.

Prime Factors

Prime factors are the building blocks of numbers. As their name suggests, these factors are prime, meaning that each one can only be divided evenly by 1 and itself without leaving a remainder. Identifying the prime factors of a number help us understand its structure and is essential for various applications in mathematics, such as finding the greatest common factor (GCF), least common multiple (LCM), or simplifying fractions.

To find the prime factors of a number, we typically use a method called prime factorization. This method involves dividing the number by the smallest prime number possible and repeating the process with the quotient until all remaining factors are prime. For example, the prime factorization of 49 is 7 × 7, since 7 is a prime number and is the only factor of 49 aside from 1.

Math Problem Solving

Math problem solving is a critical skill that extends beyond the realm of numbers and into everyday life. It involves understanding the problem at hand, devising a plan to solve it, carrying out that plan, and then looking back to review the solution. This process can be applied to a wide range of mathematical concepts, including finding the greatest common factor.

When solving math problems, it's important to apply a systematic approach to avoid confusion and to ensure accuracy. For example, in finding the greatest common factor of two numbers, you should first break down each number into its prime factors, compare these factors to identify the common ones, and determine the product of the shared factors. By doing so, you establish a clear path to the solution, which in the case of 49 and 64, leads to the discovery that their GCF is 1, as they have no common prime factors.