Terms With Greatest Common Factor Of 5m²n²

Before we dive into the specific problem, let's make sure we have a solid grasp of what the Greatest Common Factor (GCF) actually means. In the realm of mathematics, the GCF, also known as the Highest Common Factor (HCF), is the largest number or expression that divides evenly into two or more numbers or terms. Think of it as the biggest piece you can cut out of several different pies, where each piece is a whole slice. Finding the GCF is a fundamental skill in algebra and number theory, with applications ranging from simplifying fractions to solving complex equations. Understanding the GCF allows us to break down expressions into their most basic components, making them easier to work with and understand. For instance, when simplifying fractions, identifying the GCF of the numerator and denominator allows us to reduce the fraction to its simplest form. In polynomial expressions, finding the GCF can help us factorize them, which is a crucial step in solving equations and analyzing functions. The GCF isn't just a theoretical concept; it's a practical tool that streamlines mathematical operations and provides insights into the relationships between numbers and expressions. To truly master the GCF, it's essential to practice identifying it in various contexts, from simple sets of numbers to more complex algebraic expressions. This skill will not only help you solve problems more efficiently but also deepen your understanding of mathematical principles. By focusing on the underlying concept and its applications, you can build a solid foundation for more advanced mathematical concepts. So, let's keep exploring and unraveling the mysteries of the GCF, one step at a time. Remember, the GCF is more than just a number; it's a key to unlocking the hidden structures within mathematical expressions.

In our given problem, we're tasked with identifying terms that share a greatest common factor (GCF) with the expression $5m²ⁿ2$. To effectively tackle this, we need to first deconstruct $5m²ⁿ2$ itself. This expression is composed of three distinct parts: the coefficient 5, the variable m raised to the power of 2 ($m^2$), and the variable n also raised to the power of 2 ($n^2$). The coefficient 5 is a prime number, meaning its only factors are 1 and itself. This is an important observation because it limits the potential GCFs to either 1 or 5 concerning the numerical part of the terms we'll be comparing it with. Next, let's consider the variable components. The term $m^2$ represents m multiplied by itself (m * m*), and similarly, $n^2$ represents n multiplied by itself (n * n*). When determining the GCF, the exponents play a crucial role. The GCF can only contain powers of m and n that are less than or equal to the powers in $5m²ⁿ2$. This means that any term with a power of m greater than 2 or a power of n greater than 2 cannot have $5m²ⁿ2$ as its GCF. For instance, a term like $m^3$ would not work because it has a higher power of m. Understanding this breakdown is crucial for identifying which of the provided options could share a GCF with $5m²ⁿ2$. We need to look for terms that have a coefficient that shares a factor with 5 (either 1 or 5) and have powers of m and n that are less than or equal to 2. This careful consideration of each component of $5m²ⁿ2$ will guide us to the correct answers. By dissecting the expression in this manner, we gain a clearer understanding of its composition and the constraints it imposes on potential GCFs. This approach not only helps in solving this specific problem but also builds a stronger foundation for tackling more complex GCF problems in the future.

Now that we've thoroughly dissected $5m²ⁿ2$, let's turn our attention to the options provided and meticulously analyze each one to determine if it could share a greatest common factor (GCF) with our target expression. This process involves examining both the numerical coefficients and the variable exponents in each option. The first option is $m⁵ⁿ5$. While this term contains the same variables, m and n, as $5m²ⁿ2$, the exponents are significantly higher (5 for both m and n). Recall that the GCF cannot have exponents higher than those in the original term. Therefore, $m⁵ⁿ5$ cannot have $5m²ⁿ2$ as its GCF because the powers of m and n are greater than 2. Next, we consider $5m⁴ⁿ3$. This option has the same coefficient, 5, which is promising. However, the exponents on the variables are m to the power of 4 and n to the power of 3. Again, these exponents are higher than the powers in $5m²ⁿ2$, so this option also cannot have $5m²ⁿ2$ as its GCF. Moving on to $10m^4n$, we see a coefficient of 10. The factors of 10 are 1, 2, 5, and 10. This means 10 shares a factor of 5 with the coefficient of our target expression. However, the exponent of m is 4, which is greater than 2, and the exponent of n is 1, which is less than 2. Since the exponent of m is too high, this option is not a valid choice. Now, let's examine $15m²ⁿ2$. The coefficient 15 has factors 1, 3, 5, and 15, sharing the factor 5 with our target expression. Crucially, the exponents of m and n are both 2, which are the same as in $5m²ⁿ2$. This means $15m²ⁿ2$ could have $5m²ⁿ2$ as its GCF. Finally, we have $24m³ⁿ4$. The coefficient 24 has factors including 1, 2, 3, 4, 6, 8, 12, and 24. It does not share a factor of 5. Additionally, the exponents of m and n (3 and 4, respectively) are higher than those in $5m²ⁿ2$. Thus, this option is not a valid choice. By systematically analyzing each option in this way, we can confidently identify the terms that could share $5m²ⁿ2$ as their GCF. This methodical approach is essential for accurately solving GCF problems.

After a detailed analysis of each option, we're now equipped to pinpoint the terms that could indeed have $5m²ⁿ2$ as their greatest common factor (GCF). Recall that for a term to share $5m²ⁿ2$ as its GCF, it must satisfy two key conditions: first, its coefficient must share a factor with 5 (either 1 or 5), and second, the exponents of its variables m and n must be less than or equal to 2. Let's revisit our options in light of these criteria. We determined that $m⁵ⁿ5$, $5m⁴ⁿ3$, and $24m³ⁿ4$ are not viable candidates because they either have exponents greater than 2 or do not share a common factor of 5 in their coefficients. The option $10m^4n$ also fell short because, while its coefficient 10 shares a factor of 5, the exponent of m (4) exceeds the exponent of m in $5m²ⁿ2$. This leaves us with $15m²ⁿ2$. This term's coefficient, 15, has 5 as a factor, and both m and n are raised to the power of 2, satisfying our conditions perfectly. Therefore, $15m²ⁿ2$ is one of the correct options. Now, let's consider the remaining options. As we've already discussed, the exponents of m and n in $10m^4n$ don't work, so that option is out. However, upon closer inspection, the option $5m⁴ⁿ3$ is not the answer because the exponents of m and n are greater than 2. The option $15m²ⁿ2$ has a GCF of $5m²ⁿ2$. Thus, the correct options are $5m⁴ⁿ3$ and $15m²ⁿ2$. The reasoning behind this conclusion highlights the importance of a systematic approach when tackling GCF problems. By meticulously examining each component of the expressions—the coefficients and the variable exponents—we can confidently arrive at the correct solution. This process not only solves the immediate problem but also reinforces our understanding of the underlying principles of GCF, setting the stage for success with more complex algebraic challenges.

Therefore, the two options that could have a greatest common factor of $5m²ⁿ2$ are:

• $5m⁴ⁿ3$
• $15m²ⁿ2$