Problem 21 - Fundamentals of Mathematics

Question

Find the greatest common factor (GCF) of the numbers. \(60,140,\) and 100

Step-by-Step Solution

Answer

The GCF of 60, 140, and 100 is 20.

Step 1: List the Factors

First, list down all the factors of each number. For 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. For 140: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140. For 100: 1, 2, 4, 5, 10, 20, 25, 50, 100.

Step 2: Identify Common Factors

Next, identify the common factors that are present in the factors of 60, 140, and 100. Common factors: 1, 2, 4, 5, 10, 20.

Step 3: Find the Greatest Common Factor

From the list of common factors, identify the greatest factor. Look at the common factors: 1, 2, 4, 5, 10, and 20. The greatest of these is 20.

Key Concepts

Factors

Factors are numbers that divide another number without leaving a remainder. For example, if we have the number 60, its factors are all the numbers that can multiply together to make 60.
Here's a simple way to identify factors:

• If a number can be divided evenly by another number (meaning no remainder), then that other number is a factor.
• When listing factors, always start with 1 and the number itself, then move through smaller numbers to see which can divide evenly.
• For 60, the factors are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

Remember: factors come in pairs, as they multiply to the original number. This is crucial for solving problems involving greatest common factors (GCF).

Common Factors

Common factors are numbers that are factors of two or more numbers at the same time. Finding them is essential in problems requiring simplification or computation of the greatest common factor.
Here’s how you can determine them:

• List all factors for each number involved.
• Identify which factors appear in each list.
• The numbers that are present in all lists are the common factors.

Take numbers 60, 140, and 100 for example:

• The common factors here are: 1, 2, 4, 5, 10, and 20.

These common factors are smaller parts of each number that can all divide evenly. Finding these simplifies calculating the greatest common factor.

Numerical Reasoning

Numerical reasoning involves using logic and mathematical concepts to solve problems. This kind of reasoning is key in determining the greatest common factor (GCF).
Here’s a step-by-step approach to enhance your numerical reasoning skills:

• Begin by listing all factors of the given numbers, just as a way to organize what's happening numerically.
• Use comparisons to weed out the factors that different numbers share. This comparison ensures you're analyzing each number's numerical structure.
• Then, pinpoint the biggest among the shared factors, providing you with the GCF.

By following these steps, like identifying the common factors 1, 2, 4, 5, 10, and 20 for numbers 60, 140, and 100, you apply numerical reasoning effectively.
The greatest common factor in this scenario is 20, showcasing how numerical reasoning helps simplify complex problems into understandable parts.