Problem 17 - Fundamentals of Mathematics

Question

Find the greatest common factor (GCF) of the numbers. 65 and 15

Step-by-Step Solution

Answer

The GCF of 65 and 15 is 5.

Step 1: List the Factors

First, find all the factors of 65 and 15. Factors of 65: 1, 5, 13, 65. Factors of 15: 1, 3, 5, 15.

Step 2: Identify the Common Factors

Compare the lists of factors and identify which numbers are common in both lists. Common factors of 65 and 15: 1, 5.

Step 3: Choose the Greatest Factor

Among the common factors, select the greatest number. The greatest common factor from 1 and 5 is 5.

Key Concepts

Factors

To find the greatest common factor, it's important to first understand what factors are. Factors are numbers that divide another number without leaving a remainder.
For example, the factors of 65, as mentioned in the problem, are 1, 5, 13, and 65. This means that each of these numbers divides 65 perfectly. Similarly, the factors of 15 are 1, 3, 5, and 15.
Here's how you can find factors of any number:
- Start with the number 1 (as it is a factor of every whole number) and the number itself.
- Then, check every integer between 1 and the number to see if it divides the number without a remainder.
Finding factors is the foundational step in identifying the greatest common factor, or GCF.

Common Factors

Once the factors of each number are known, the next step is to identify the common factors. These are the factors that appear in both lists of factors.
For the numbers 65 and 15, the two lists contain the numbers 1 and 5. These are the common factors because they appear in both lists.
To identify common factors:
- List all factors of the first number.
- List all factors of the second number.
- Compare the lists and find the numbers that appear in both.
Identifying common factors helps narrow down the possibilities to find the greatest common factor, making it an essential skill in both elementary arithmetic and more advanced math applications.

Arithmetic

Arithmetic is the branch of mathematics dealing with numbers and simple operations such as addition, subtraction, multiplication, and division.
When we calculate factors, we use basic arithmetic skills.
Calculating the greatest common factor involves these arithmetic operations:
- Identifying factors through division.
- Comparing numbers to find common elements.
- Selecting the greatest of these common elements.
The process uses arithmetic to break down numbers into their simplest components and understand their relationships. Understanding these operations and their applications can make solving problems involving factors more intuitive and less daunting. Mastery of arithmetic lays the groundwork for tackling more complex mathematical problems in the future.