Prime Factorization of 3825

Oct 13, 2025 by miodeya.com 27 views

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Introduction

The given problem involves finding the value of x + y - 2z in the equation 3825 = 3^x × 5^y × 17^z. To solve this problem, we need to first find the prime factorization of 3825. Once we have the prime factorization, we can compare it with the given equation and find the values of x, y, and z.

Prime Factorization of 3825

To find the prime factorization of 3825, we can start by dividing it by the smallest prime number, which is 2. However, 3825 is an odd number, so we can start by dividing it by 3.

3825 ÷ 3 = 1275

Now, we can divide 1275 by 3 again.

1275 ÷ 3 = 425

425 is not divisible by 3, so we can try dividing it by 5.

425 ÷ 5 = 85

85 is not divisible by 5, so we can try dividing it by 17.

85 ÷ 17 = 5

Now, we have the prime factorization of 3825 as 3^4 × 5^2 × 17^1.

Comparing with the Given Equation

Now that we have the prime factorization of 3825, we can compare it with the given equation 3825 = 3^x × 5^y × 17^z.

From the prime factorization, we can see that x = 4, y = 2, and z = 1.

Finding the Value of x + y - 2z

Now that we have the values of x, y, and z, we can find the value of x + y - 2z.

x + y - 2z = 4 + 2 - 2(1) = 4 + 2 - 2 = 6 - 2 = 4

Therefore, the value of x + y - 2z is 4.

Conclusion

In this problem, we were given the equation 3825 = 3^x × 5^y × 17^z and asked to find the value of x + y - 2z. To solve this problem, we first found the prime factorization of 3825, which is 3^4 × 5^2 × 17^1. Then, we compared it with the given equation and found the values of x, y, and z. Finally, we found the value of x + y - 2z, which is 4.

Tips and Tricks

To solve this problem, you need to have a good understanding of prime factorization and how to compare it with a given equation.
Make sure to read the problem carefully and understand what is being asked.
Use the prime factorization to find the values of x, y, and z, and then use those values to find the value of x + y - 2z.

Frequently Asked Questions

What is the prime factorization of 3825?
- The prime factorization of 3825 is 3^4 × 5^2 × 17^1.
How do I find the value of x + y - 2z?
- To find the value of x + y - 2z, you need to first find the values of x, y, and z by comparing the prime factorization with the given equation. Then, you can use those values to find the value of x + y - 2z.

Q: What is the prime factorization of 3825?

A: The prime factorization of 3825 is 3^4 × 5^2 × 17^1.

Q: How do I find the value of x + y - 2z?

A: To find the value of x + y - 2z, you need to first find the values of x, y, and z by comparing the prime factorization with the given equation. Then, you can use those values to find the value of x + y - 2z.

Q: What is the value of x + y - 2z?

A: The value of x + y - 2z is 4.

Q: How do I compare the prime factorization with the given equation?

A: To compare the prime factorization with the given equation, you need to match the powers of the prime factors with the given equation. In this case, the prime factorization is 3^4 × 5^2 × 17^1, and the given equation is 3825 = 3^x × 5^y × 17^z. By matching the powers, you can find the values of x, y, and z.

Q: What if the prime factorization is not given?

A: If the prime factorization is not given, you can try to find it by dividing the number by the smallest prime numbers. In this case, we started by dividing 3825 by 3, and then continued dividing by 3, 5, and 17 until we reached the prime factorization.

Q: How do I find the value of x, y, and z?

A: To find the value of x, y, and z, you need to compare the prime factorization with the given equation. By matching the powers of the prime factors, you can find the values of x, y, and z.

Q: What if the values of x, y, and z are not integers?

A: If the values of x, y, and z are not integers, you may need to use a different method to find the value of x + y - 2z. However, in this case, the values of x, y, and z are integers, so we can use the method described above.

Q: Can I use a calculator to find the value of x + y - 2z?

A: Yes, you can use a calculator to find the value of x + y - 2z. However, it's always a good idea to understand the underlying math and how to solve the problem manually.

Q: How do I apply this method to other problems?

A: To apply this method to other problems, you need to follow the same steps:

Find the prime factorization of the number.
Compare the prime factorization with the given equation.
Find the values of x, y, and z by matching the powers of the prime factors.
Use the values of x, y, and z to find the value of x + y - 2z.

Q: What are some common mistakes to avoid?

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

Not finding the prime factorization correctly.
Not comparing the prime factorization with the given equation correctly.
Not finding the values of x, y, and z correctly.
Not using the correct method to find the value of x + y - 2z.

Q: How do I practice and improve my skills?

A: To practice and improve your skills, you can try solving similar problems and exercises. You can also ask your teacher or tutor for help and guidance.

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

A: This method has many real-world applications, including:

Cryptography: This method is used in cryptography to encode and decode messages.
Coding theory: This method is used in coding theory to design error-correcting codes.
Number theory: This method is used in number theory to study the properties of numbers.

Q: Can I use this method to solve other types of problems?

A: Yes, you can use this method to solve other types of problems that involve prime factorization and comparison with a given equation. However, the specific method and steps may vary depending on the problem.