Problem 93 - 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 6 and 10 is 2.

Step-by-Step Solution

Answer

The statement 'The greatest common factor of 6 and 10 is 2.' is true.

Step 1: Understand the 'greatest common factor'

The 'greatest common factor' (GCF), also known as the 'greatest common divisor' (GCD), is the largest natural number that divides two numbers without leaving a remainder. In this case, we're looking for the GCF of 6 and 10.

Step 2: List Factors of Each Number

First, identify the factors of each number. Factors of 6 include 1, 2, 3, and 6. Factors of 10 include 1, 2, 5, 10. Look for the common factors of these two numbers, which are 1 and 2.

Step 3: Identify the Greatest Common Factor

The common factors between 6 and 10 are 1 and 2. Out of these, 2 is the greatest.

Step 4: Evaluate the Statement

The statement given in the exercise is 'The greatest common factor of 6 and 10 is 2.' Since we have identified 2 as the greatest common factor of 6 and 10, the statement is true.

Key Concepts

GCF

The term 'Greatest Common Factor' (GCF) is a cornerstone in the domain of number theory and plays a significant role in various mathematical applications. At its essence, GCF identifies the largest number that evenly divides two or more numbers without leaving a remainder. For instance, when deciphering the GCF of 6 and 10, we're essentially searching for the biggest number that divides into both 6 and 10 neatly.

Calculating the GCF can be approached in a few different ways, such as listing out all the factors of the numbers involved, or employing the Euclidean algorithm, which is a more efficient method, particularly for larger integers. The GCF is widely utilized in simplifying fractions, finding common denominators, and solving problems that involve ratios or proportions.

Factors of Numbers

Factors of a number are all the whole numbers that can multiply together to yield the targeted number. To determine the factors of a number, you'd iterate through integers from 1 up to the number itself and check their divisibility. For example, the number 6 has factors of 1, 2, 3, and 6 itself—because 1 x 6 = 6 and 2 x 3 = 6.

When comparing the factors of 6 (1, 2, 3, 6) with those of 10 (1, 2, 5, 10), the common factors are the ones present in both lists, 1 and 2 in this case. Understanding factors is crucial for various branches of mathematics, including algebra and number theory, as it's fundamental in finding multiples, divisibility, and performing operations with fractions.

Divisibility

Divisibility is a concept that refers to the ability of one number to be divided by another without leaving a remainder. It's a simple test to gauge whether a division operation between two integers will result in an integer. Common rules of divisibility can swiftly determine if a number is divisible by another, such as even numbers are always divisible by 2 or any number ending in 0 or 5 is divisible by 5.

To illustrate, take the number 10; we know immediately it is divisible by 2 because it's even, while the certainty that it's divisible by 5 comes from it ending with a 0. Divisibility tests streamline the process of finding factors and GCF. They are also essential for simplifying mathematical expressions and understanding the properties of integers.