Problem 65 - Fundamentals of Mathematics

Question

Find the greatest common factor of each collection of numbers. 5 and 15

Step-by-Step Solution

Answer

The greatest common factor of 5 and 15 is 5.

Step 1: Understand the Concept

The greatest common factor (GCF) of a set of numbers is the largest number that divides all of them without leaving a remainder. In other words, it's the biggest number that all the numbers in the set have in common as a factor.

Step 2: List the Factors of Each Number

Begin by listing all factors of each number. - Factors of 5: 1, 5. - Factors of 15: 1, 3, 5, 15.

Step 3: Identify the Common Factors

Look at the lists of factors from Step 2 and identify which ones the numbers have in common. - Common factors of 5 and 15: 1, 5.

Step 4: Determine the Greatest Common Factor

From the common factors identified in Step 3, find the greatest one. - The greatest common factor of 5 and 15 is 5.

Key Concepts

Factors

Factors are numbers that divide a given number completely without leaving any remainder. For any number, the factors include the number itself and 1, along with any other numbers that fit this criterion. For example, if we look at the number 15, its factors are 1, 3, 5, and 15. These numbers can multiply with something to produce 15 without any leftovers.

To find the factors of a number:

• Start with 1, which is a factor for every number.
• Check each number up to the given number itself to see if it divides evenly without a remainder.

Learning to quickly identify factors will help immensely in solving other math problems like finding greatest common factors or simplifying fractions.

Common Factors

Common factors are factors that two or more numbers share. These are the numbers that appear in the factor lists of each number you are considering. For example, if we take the numbers 5 and 15, the common factors from their individual factor lists (1, 5 and 1, 3, 5, 15) are 1 and 5.

Finding common factors involves:

• Listing all factors of each number individually.
• Identifying the numbers that appear in all lists.

Common factors are especially important when calculating the greatest common factor (GCF), which is derived from the largest shared factor among numbers. Recognizing these shared factors plays a crucial role in solving problems related to divisibility and simplifying expressions.

Divisibility

Divisibility refers to how a number can be divided by another number without leaving a remainder. Understanding divisibility is key when discovering factors and common factors. For a number to be a factor, it must be divisible by your target number, meaning it divides fully without any leftover.

When determining divisibility:

• Review basic divisibility rules like any number is divisible by 1, numbers ending in an even number are divisible by 2, and so on.
• Apply these rules to quickly identify factors without long division.

Divisibility helps streamline the process for finding the greatest common factor. Comprehending these simple rules aids students in solving problems across mathematics.