Problem 18 - Fundamentals of Mathematics

Question

Find the greatest common factor (GCF) of the numbers. 33 and 77

Step-by-Step Solution

Answer

The GCF of 33 and 77 is 11.

Step 1: List Factors of Each Number

The first step is to determine the factors of each number. Factors of a number are integers that can be multiplied together to equal the original number. - The factors of 33 are 1, 3, 11, and 33. - The factors of 77 are 1, 7, 11, and 77.

Step 2: Identify Common Factors

Next, we identify any factors that are common to both numbers. Looking at our list of factors: - Common factors of 33 and 77 are 1 and 11.

Step 3: Determine the GCF

The greatest common factor (GCF) is the largest factor that both numbers share. From the common factors identified (1 and 11), the greatest factor is 11.

Key Concepts

Factors of a Number

Factors of a number are simply numbers that, when multiplied together, give you the original number. Imagine you have a puzzle, and your task is to find pieces that fit perfectly together without any gaps. Factors work in the same way, as they "fit" with the number perfectly when multiplied. For instance:

• For the number 33, the factors are 1, 3, 11, and 33.
• For the number 77, the factors are 1, 7, 11, and 77.

Each factor is a whole number, and every number will include at least 1 and itself as factors. Being able to list requirements is essential in identifying common factors, which we will discuss next.

Common Factors

Common factors are factors that two or more numbers share. Think of it like finding common friends; these are groups of numbers that both can "hang out with" equally. To identify these, look at the list of factors you've determined, such as:

• For 33: 1, 3, 11, 33
• For 77: 1, 7, 11, 77

Now, see which numbers they have in common. Here, both 33 and 77 share 1 and 11 as their common factors. The concept of commonality helps in many areas of math, especially when simplifying fractions or understanding relationships between numbers.

Finding GCF

Finding the GCF, or greatest common factor, involves determining the largest number from the common factors. Think of it as finding the most significant "friend" the numbers have in common. From our example, both numbers 33 and 77 have common factors: 1 and 11. Among these, 11 is the largest, making it the greatest common factor.

• GCF helps when reducing fractions, as it is the key number that you use to simplify both numerator and denominator.
• Understanding GCF can simplify problems involving ratios, proportions, and even some equations.

By mastering how to determine the GCF, you enhance your mathematical problem-solving skills and make complex problems much easier to manage.