Problem 97 - Algebra 1: Concepts and Skills

Question

Decide whether the statement is true or false. If it is false, correct the statement to make it true. The greatest common factor of 45 and 82 is 3.

Step-by-Step Solution

Answer

The initial statement is false. The greatest common factor of 45 and 82 is 1, not 3.

Step 1: Identifying the problem

The problem is to determine if 3 is the greatest common factor of 45 and 82. This needs verification by identifying the factors of both 45 and 82.

Step 2: Finding the factors of 45 and 82

The factors of 45 are: 1, 3, 5, 9, 15, 45. The factors of 82 are: 1, 2, 41, 82.

Step 3: Identify the greatest common factor of 45 and 82

Using the list of factors from the previous step, it's clear that 1 is the only factor common to both 45 and 82. Therefore, 1 is the greatest common factor of 45 and 82, not 3 as the initial statement indicated.

Key Concepts

Factors of Numbers

Understanding the factors of numbers is essential in mathematics, particularly when solving problems related to divisibility. A factor of a number is one that can be multiplied by another integer to yield the original number. Factors can be positive or negative, but we usually focus on positive integers when discussing the subject. For example, the numbers that divide 45 without leaving any remainder are its factors. These include 1, 3, 5, 9, 15, and 45. Each of these numbers can multiply with another integer to give you 45. Similarly, for 82, the factors are 1, 2, 41, and 82. To find factors, you can start from 1 and work your way up to the number itself. For instance, when finding the factors of 45, begin by checking divisibility by 1, which is a universal factor, then move on to 2, 3, and so on. Remember that efficient factorization is a useful skill that aids in many mathematical processes, such as finding the greatest common factor or simplifying fractions.

Common Factors

Common factors are numbers that are shared as factors between two or more numbers. Discovering common factors between numbers is straightforward: simply list the factors of each number and identify those that appear in each list. For example, between 45 and 82, both lists of factors include the number 1, making it a common factor. The greatest common factor (GCF) is the largest factor that the numbers share. It's crucial for simplifying fractions or solving problems involving ratios. In the case of 45 and 82, the only common factor is 1, meaning the GCF is also 1. Always double-check each factor list to ensure no factors are overlooked. Understanding common and greatest common factors can significantly streamline problem-solving in arithmetic and algebra.

Mathematics Problem Solving

Problem-solving is a central skill in mathematics that requires logical thinking and a systematic approach. When tackling problems like finding the greatest common factor, it’s essential to proceed methodically:

• Understand the problem statement clearly. For instance, if you're asked to validate a claim about the greatest common factor, know the definitions involved.
• List out all relevant factors of the numbers involved. As seen with 45 and 82, identifying these lists is critical.
• Compare the lists to find common factors. Highlight any commonality, no matter how small, like the number 1 being a factor of both 45 and 82.

After identifying common factors, sift through them to confirm the greatest one. This process ensures accuracy and frees students from common misconceptions, such as misidentifying non-factors or mistaking a relative number for a factor. Regular practice of these steps will bolster problem-solving skills and increase confidence when dealing with complex mathematical scenarios.