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Chapter 10: Problem 423

Find the greatest common factor. $$45,75$$

Short Answer

Expert verified
The greatest common factor of 45 and 75 is 15.

Step by step solution

01

List the factors of the first number

List all the factors of 45. These are: 1, 3, 5, 9, 15, 45.
02

List the factors of the second number

List all the factors of 75. These are: 1, 3, 5, 15, 25, 75.
03

Find the common factors

Identify the common factors from both lists. The common factors of 45 and 75 are: 1, 3, 5, 15.
04

Determine the greatest common factor

From the list of common factors, select the greatest one. The greatest common factor (GCF) of 45 and 75 is 15.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factors
Factors are numbers that can be multiplied together to get another number. To find the factors of a number, we start by testing each whole number to see if it divides evenly into the original number.
For example, the factors of 45 are the numbers that can be multiplied in pairs to get 45. These pairs are:
  • 1 脳 45
  • 3 脳 15
  • 5 脳 9
So, the factors of 45 are: 1, 3, 5, 9, 15, 45.
Similarly, the factors of 75 are:
  • 1 脳 75
  • 3 脳 25
  • 5 脳 15
Leading to the factors of 75: 1, 3, 5, 15, 25, 75.
Number Theory
Number theory is a branch of mathematics concerned with the properties of numbers, especially integers. It includes various concepts like divisibility, prime numbers, and common factors.
Understanding factors and how they work is a fundamental part of number theory.
In the context of finding the greatest common factor (GCF), we use number theory to break down each number into its factors and then look for shared factors.
This involves testing which numbers divide evenly into the initial numbers and identifying the biggest number that can divide both numbers without leaving a remainder.
GCF Calculation
The greatest common factor (GCF) is the largest number that can divide two or more numbers without leaving a remainder. To calculate the GCF, we follow these steps:
  1. List down all factors of the first number.
  2. List down all factors of the second number.
  3. Identify the common factors between the two lists.
  4. Select the largest of these common factors.
In our example, we listed the factors of 45: 1, 3, 5, 9, 15, 45.
We also listed the factors of 75: 1, 3, 5, 15, 25, 75.
The common factors here are: 1, 3, 5, 15. Finally, we select the largest common factor, which is 15.
Hence, the GCF of 45 and 75 is 15.

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