Four Students Were Asked To Write An Expression Which Has Terms That Have A Greatest Common Factor Of A B A B Ab . The Expressions Provided By The Students Are Shown Below.$[ \begin{tabular}{|c|c|} \hline \text{Michelle} & \text{Naomi} \ 16a +

Mar 10, 2025 by ADMIN 244 views

Four Students' Expressions with Greatest Common Factor

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Introduction

In mathematics, the greatest common factor (GCF) of a set of numbers is the largest positive integer that divides each of the numbers without leaving a remainder. In this article, we will explore the expressions provided by four students, each with terms that have a greatest common factor of $a b$ . We will analyze and compare their expressions to understand the concept of GCF and its application in algebra.

Michelle's Expression

Michelle's expression is $16a + 8b$ . To find the greatest common factor of this expression, we need to identify the largest positive integer that divides both $16a$ and $8b$ without leaving a remainder.

import math
def find_gcf(a, b):
return math.gcd(a, b)
a = 16
b = 8
gcf = find_gcf(a, b)
print("The greatest common factor of", a, "and", b, "is", gcf)

The greatest common factor of $16a$ and $8b$ is $8$ . Therefore, Michelle's expression can be rewritten as $8(2a + b)$ .

Naomi's Expression

Naomi's expression is $16a + 24b$ . To find the greatest common factor of this expression, we need to identify the largest positive integer that divides both $16a$ and $24b$ without leaving a remainder.

import math
def find_gcf(a, b):
return math.gcd(a, b)
a = 16
b = 24
gcf = find_gcf(a, b)
print("The greatest common factor of", a, "and", b, "is", gcf)

The greatest common factor of $16a$ and $24b$ is $8$ . Therefore, Naomi's expression can be rewritten as $8(2a + 3b)$ .

Comparison of Expressions

Let's compare Michelle's and Naomi's expressions:

Michelle's expression: $8(2a + b)$
Naomi's expression: $8(2a + 3b)$

We can see that both expressions have a greatest common factor of $8$ . However, the coefficients of $a$ and $b$ are different in each expression.

Other Students' Expressions

Two other students, Alex and Ben, also provided expressions with terms that have a greatest common factor of $a b$ . Alex's expression is $24a + 16b$ , and Ben's expression is $32a + 20b$ .

import math
def find_gcf(a, b):
return math.gcd(a, b)
a = 24
b = 16
gcf = find_gcf(a, b)
print("The greatest common factor of", a, "and", b, "is", gcf)

a = 32
b = 20
gcf = find_gcf(a, b)
print("The greatest common factor of", a, "and", b, "is", gcf)

The greatest common factor of $24a$ and $16b$ is $8$ . The greatest common factor of $32a$ and $20b$ is $4$ .

Conclusion

In this article, we analyzed the expressions provided by four students, each with terms that have a greatest common factor of $a b$ . We found that Michelle's and Naomi's expressions can be rewritten as $8(2a + b)$ and $8(2a + 3b)$ , respectively. We also compared their expressions and found that the coefficients of $a$ and $b$ are different in each expression. Additionally, we analyzed the expressions provided by Alex and Ben and found that the greatest common factor of $24a$ and $16b$ is $8$ , and the greatest common factor of $32a$ and $20b$ is $4$ .

Applications of Greatest Common Factor

The greatest common factor has many applications in mathematics and real-world problems. Some examples include:

Finding the greatest common divisor of two numbers
Simplifying fractions
Factoring polynomials
Solving systems of linear equations

Final Thoughts

In conclusion, the greatest common factor is an important concept in mathematics that has many applications in real-world problems. By understanding the concept of GCF, we can simplify expressions, solve systems of linear equations, and find the greatest common divisor of two numbers. We hope that this article has provided a clear understanding of the concept of GCF and its applications in mathematics.

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Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) of a set of numbers is the largest positive integer that divides each of the numbers without leaving a remainder.

Q: How do I find the greatest common factor of two numbers?

A: To find the greatest common factor of two numbers, you can use the following steps:

List the factors of each number.
Identify the common factors.
Choose the largest common factor.

Alternatively, you can use the Euclidean algorithm to find the GCF.

Q: What is the Euclidean algorithm?

A: The Euclidean algorithm is a method for finding the greatest common factor of two numbers. It works by repeatedly dividing the larger number by the smaller number and taking the remainder.

Q: How do I use the Euclidean algorithm to find the GCF?

A: Here's an example of how to use the Euclidean algorithm to find the GCF of 48 and 18:

Divide 48 by 18: 48 = 2(18) + 12
Divide 18 by 12: 18 = 1(12) + 6
Divide 12 by 6: 12 = 2(6) + 0

The last non-zero remainder is 6, so the GCF of 48 and 18 is 6.

Q: What is the difference between GCF and LCM?

A: The greatest common factor (GCF) and least common multiple (LCM) are two related but distinct concepts.

The GCF is the largest positive integer that divides each of the numbers without leaving a remainder.
The LCM is the smallest positive integer that is a multiple of each of the numbers.

Q: How do I find the least common multiple (LCM) of two numbers?

A: To find the least common multiple (LCM) of two numbers, you can use the following steps:

List the multiples of each number.
Identify the smallest common multiple.

Alternatively, you can use the formula:

LCM(a, b) = (a × b) / GCF(a, b)

Q: What are some real-world applications of GCF and LCM?

A: GCF and LCM have many real-world applications, including:

Simplifying fractions
Factoring polynomials
Solving systems of linear equations
Finding the greatest common divisor of two numbers
Finding the least common multiple of two numbers

Q: Can I use a calculator to find the GCF and LCM?

A: Yes, you can use a calculator to find the GCF and LCM. Most calculators have a built-in function for finding the GCF and LCM.

Q: What are some common mistakes to avoid when finding the GCF and LCM?

A: Some common mistakes to avoid when finding the GCF and LCM include:

Not listing all the factors of each number
Not identifying the common factors
Not choosing the largest common factor
Not using the Euclidean algorithm correctly

Q: Can I find the GCF and LCM of more than two numbers?

A: Yes, you can find the GCF and LCM of more than two numbers. To do this, you can use the following steps:

Find the GCF of the first two numbers
Find the GCF of the result and the third number
Continue this process until you have found the GCF of all the numbers

Alternatively, you can use the formula:

GCF(a, b, c) = GCF(GCF(a, b), c)

LCM(a, b, c) = LCM(LCM(a, b), c)

Q: What are some advanced topics related to GCF and LCM?

A: Some advanced topics related to GCF and LCM include:

Prime factorization
Greatest common divisor of polynomials
Least common multiple of polynomials
GCD and LCM of complex numbers

Q: Can I use GCF and LCM to solve real-world problems?

A: Yes, you can use GCF and LCM to solve real-world problems. Some examples include:

Simplifying fractions in cooking recipes
Factoring polynomials in engineering design
Solving systems of linear equations in physics
Finding the greatest common divisor of two numbers in cryptography

Q: What are some resources for learning more about GCF and LCM?

A: Some resources for learning more about GCF and LCM include:

Online tutorials and videos
Math textbooks and workbooks
Online communities and forums
Math apps and software

Q: Can I use GCF and LCM to solve problems in other subjects?

A: Yes, you can use GCF and LCM to solve problems in other subjects, including:

Science: to solve systems of linear equations and find the greatest common divisor of two numbers
Engineering: to factor polynomials and find the least common multiple of two numbers
Computer Science: to find the greatest common divisor of two numbers and solve systems of linear equations

Q: What are some common misconceptions about GCF and LCM?

A: Some common misconceptions about GCF and LCM include:

Thinking that GCF and LCM are the same thing
Thinking that GCF and LCM are only used in math class
Thinking that GCF and LCM are only used to solve simple problems

Q: Can I use GCF and LCM to solve problems in real-time?

A: Yes, you can use GCF and LCM to solve problems in real-time. Some examples include:

Using a calculator to find the GCF and LCM of two numbers in a math competition
Using a computer program to find the GCF and LCM of two numbers in a science experiment
Using a math app to find the GCF and LCM of two numbers in a real-world problem.