Find Number Of Solutions Of Equation X Y Z W = 2 5 ⋅ 3 7 ⋅ 5 4 Xyzw=2^{5}\cdot 3^7 \cdot 5^{4} X Yz W = 2 5 ⋅ 3 7 ⋅ 5 4 , Where X , Y , Z , W ∈ N X,y,z,w\in \mathbb N X , Y , Z , W ∈ N Such That HCF Of X , Y , Z , W X,y,z,w X , Y , Z , W Is 1.

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Introduction

The problem of finding the number of solutions to the equation $xyzw=2^{5}\cdot 3^7 \cdot 5^{4}$, where $x,y,z,w\in \mathbb N$ such that HCF of $x,y,z,w$ is 1, is a classic problem in number theory. This problem involves the concept of greatest common divisor (GCD) and least common multiple (LCM), which are fundamental concepts in number theory.

Understanding the Problem

To solve this problem, we need to understand the concept of GCD and LCM. The GCD of two numbers is the largest number that divides both of them without leaving a remainder, while the LCM of two numbers is the smallest number that is a multiple of both of them. In this problem, we are given the equation $xyzw=2^{5}\cdot 3^7 \cdot 5^{4}$, and we need to find the number of solutions to this equation such that the HCF of $x,y,z,w$ is 1.

Using the Concept of LCM and GCD

As we know, $\text{LCM}\cdot \text{HCF}=2^{5}\cdot3^{7}\cdot5^{4}$. Since the HCF of $x,y,z,w$ is 1, we can conclude that $\text{LCM}(x,y,z,w)=2^{5}\cdot3^{7}\cdot5^{4}$. This means that the LCM of $x,y,z,w$ is equal to the product of the prime factors of the given equation.

Breaking Down the LCM

We can break down the LCM into its prime factors as follows:

$\text{LCM}(x,y,z,w)=2^{5}\cdot3^{7}\cdot5^{4}$

This means that the LCM of $x,y,z,w$ is equal to the product of the prime factors $2^{5}$, $3^{7}$, and $5^{4}$.

Finding the Number of Solutions

To find the number of solutions to the equation $xyzw=2^{5}\cdot 3^7 \cdot 5^{4}$, we need to find the number of ways to distribute the prime factors $2^{5}$, $3^{7}$, and $5^{4}$ among the variables $x$, $y$, $z$, and $w$. This is a classic problem in combinatorics, and it can be solved using the concept of combinations.

Using Combinations to Find the Number of Solutions

We can use the concept of combinations to find the number of ways to distribute the prime factors $2^{5}$, $3^{7}$, and $5^{4}$ among the variables $x$, $y$, $z$, and $w$. Let's consider the prime factor $2^{5}$ first. We can distribute this prime factor among the variables $x$, $y$, $z$, and $w$ in $4^{5}$ ways, since each variable can receive any of the $2^{5}$ prime factors.

Distributing the Prime Factor $3^{7}$

Next, we need to distribute the prime factor $3^{7}$ among the variables $x$, $y$, $z$, and $w$. We can do this in $4^{7}$ ways, since each variable can receive any of the $3^{7}$ prime factors.

Distributing the Prime Factor $5^{4}$

Finally, we need to distribute the prime factor $5^{4}$ among the variables $x$, $y$, $z$, and $w$. We can do this in $4^{4}$ ways, since each variable can receive any of the $5^{4}$ prime factors.

Finding the Total Number of Solutions

To find the total number of solutions to the equation $xyzw=2^{5}\cdot 3^7 \cdot 5^{4}$, we need to multiply the number of ways to distribute each prime factor among the variables. This gives us a total of $4^{5}\cdot 4^{7}\cdot 4^{4}$ ways to distribute the prime factors.

Simplifying the Expression

We can simplify the expression $4^{5}\cdot 4^{7}\cdot 4^{4}$ by combining the exponents. This gives us $4^{16}$.

Finding the Final Answer

Therefore, the final answer to the problem is $4^{16}$.

Conclusion

In this article, we have discussed the problem of finding the number of solutions to the equation $xyzw=2^{5}\cdot 3^7 \cdot 5^{4}$, where $x,y,z,w\in \mathbb N$ such that HCF of $x,y,z,w$ is 1. We have used the concept of LCM and GCD to solve this problem, and we have found that the total number of solutions is $4^{16}$. This problem is a classic example of a combinatorics problem, and it requires a deep understanding of the concepts of LCM and GCD.

References

• [1] "Combinatorics" by Richard P. Stanley
• [2] "Number Theory" by George E. Andrews
• [3] "Algebra" by Michael Artin

Glossary

• GCD: Greatest Common Divisor
• LCM: Least Common Multiple
• HCF: Highest Common Factor
• Combinatorics: The branch of mathematics that deals with counting and arranging objects in various ways.
• Number Theory: The branch of mathematics that deals with the properties and behavior of integers and other whole numbers.
  

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Q: What is the problem of finding the number of solutions to the equation $xyzw=2^{5}\cdot 3^7 \cdot 5^{4}$?

A: The problem is to find the number of solutions to the equation $xyzw=2^{5}\cdot 3^7 \cdot 5^{4}$, where $x,y,z,w\in \mathbb N$ such that HCF of $x,y,z,w$ is 1.

Q: What is the concept of LCM and GCD, and how are they related to this problem?

A: The LCM of two numbers is the smallest number that is a multiple of both of them, while the GCD of two numbers is the largest number that divides both of them without leaving a remainder. In this problem, we use the concept of LCM and GCD to find the number of solutions to the equation $xyzw=2^{5}\cdot 3^7 \cdot 5^{4}$.

Q: How do we break down the LCM into its prime factors?

A: We can break down the LCM into its prime factors by expressing it as a product of prime numbers. In this case, the LCM is $2^{5}\cdot 3^{7}\cdot 5^{4}$, which can be broken down into its prime factors as $2^{5}$, $3^{7}$, and $5^{4}$.

Q: How do we find the number of solutions to the equation $xyzw=2^{5}\cdot 3^7 \cdot 5^{4}$?

A: We can find the number of solutions to the equation $xyzw=2^{5}\cdot 3^7 \cdot 5^{4}$ by using the concept of combinations. We can distribute the prime factors $2^{5}$, $3^{7}$, and $5^{4}$ among the variables $x$, $y$, $z$, and $w$ in $4^{5}\cdot 4^{7}\cdot 4^{4}$ ways.

Q: What is the total number of solutions to the equation $xyzw=2^{5}\cdot 3^7 \cdot 5^{4}$?

A: The total number of solutions to the equation $xyzw=2^{5}\cdot 3^7 \cdot 5^{4}$ is $4^{16}$.

Q: What is the significance of this problem in the field of mathematics?

A: This problem is a classic example of a combinatorics problem, and it requires a deep understanding of the concepts of LCM and GCD. It is also a good example of how to use the concept of combinations to solve problems in mathematics.

Q: What are some real-world applications of this problem?

A: This problem has many real-world applications in fields such as computer science, engineering, and economics. For example, it can be used to model the behavior of complex systems, such as networks and algorithms.

Q: How can I learn more about this problem and its applications?

A: You can learn more about this problem and its applications by reading books and articles on combinatorics and number theory. You can also take online courses or attend workshops and conferences to learn more about this topic.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

• Not breaking down the LCM into its prime factors
• Not using the concept of combinations to find the number of solutions
• Not checking for errors in the calculation of the total number of solutions

Q: How can I practice solving this problem?

A: You can practice solving this problem by working on similar problems and exercises. You can also try to come up with your own problems and solutions to practice your skills.

Q: What are some resources available for learning more about this problem and its applications?

A: Some resources available for learning more about this problem and its applications include:

• Books on combinatorics and number theory
• Online courses and tutorials
• Workshops and conferences
• Research papers and articles

Q: How can I get help if I am stuck on this problem?

A: If you are stuck on this problem, you can try the following:

• Ask a teacher or tutor for help
• Search online for solutions and explanations
• Join a study group or online community to discuss the problem with others
• Try to come up with your own solution to the problem.