Determine All Three-digit Positive Integers Abc
Introduction
In this article, we will explore the problem of determining all three-digit positive integers abc (where a is the hundreds digit, b is the tens digit, and c is the ones digit) such that 8abc = 3cba. This problem falls under the category of Elementary Number Theory and Integers. We will start by analyzing the given equation and then proceed to solve for the possible values of a, b, and c.
The Given Equation
The given equation is 8abc = 3cba. We can rewrite this equation as:
8(100a + 10b + c) = 3(100c + 10b + a)
Expanding the equation, we get:
800a + 80b + 8c = 300c + 30b + 3a
Simplifying the Equation
We can simplify the equation by rearranging the terms:
800a - 3a + 80b - 30b + 8c - 300c = 0
Combining like terms, we get:
797a - 50b - 292c = 0
Dividing by the Greatest Common Divisor
The greatest common divisor (GCD) of 797, 50, and 292 is 1. Therefore, we can divide the equation by 1 to simplify it further:
797a - 50b - 292c = 0
Analyzing the Equation
We can analyze the equation by looking at the coefficients of a, b, and c. The coefficient of a is 797, which is a prime number. The coefficient of b is -50, which is a negative multiple of 10. The coefficient of c is -292, which is a negative multiple of 4.
Finding the Possible Values of a, b, and c
We can find the possible values of a, b, and c by analyzing the equation. Since the coefficient of a is 797, which is a prime number, a must be a multiple of 797. However, since a is a digit, it must be between 1 and 9. Therefore, a must be 1.
Substituting a = 1 into the equation, we get:
797 - 50b - 292c = 0
Simplifying the equation, we get:
-50b - 292c = -797
Dividing both sides by -1, we get:
50b + 292c = 797
Finding the Possible Values of b and c
We can find the possible values of b and c by analyzing the equation. Since the coefficient of b is 50, which is a multiple of 10, b must be a multiple of 10. However, since b is a digit, it must be between 0 and 9. Therefore, b must be 0 or 8.
If b = 0, then:
292c = 797
Dividing both sides by 292, we get:
c = 797/292
c = 2.73 (approximately)
Since ** is a digit, it must be an integer. Therefore, b cannot be 0.
If b = 8, then:
50(8) + 292c = 797
Simplifying the equation, we get:
400 + 292c = 797
Subtracting 400 from both sides, we get:
292c = 397
Dividing both sides by 292, we get:
c = 397/292
c = 1.36 (approximately)
Since c is a digit, it must be an integer. Therefore, c must be 1.
The Final Answer
Therefore, the only possible values of a, b, and c are:
a = 1 b = 8 c = 1
The final answer is 118.
Conclusion
Introduction
In our previous article, we determined all three-digit positive integers abc such that 8abc = 3cba. In this article, we will provide a Q&A section to answer some common questions related to this problem.
Q: What is the significance of the equation 8abc = 3cba?
A: The equation 8abc = 3cba is a classic example of a number theory problem. It involves the concept of symmetry and the properties of numbers. The equation is asking us to find all three-digit positive integers abc such that when we reverse the digits, the resulting number is a multiple of 8.
Q: Why is the coefficient of a 797?
A: The coefficient of a is 797 because it is a prime number. This means that a must be a multiple of 797. However, since a is a digit, it must be between 1 and 9. Therefore, a must be 1.
Q: Why is the coefficient of b -50?
A: The coefficient of b is -50 because it is a negative multiple of 10. This means that b must be a multiple of 10. However, since b is a digit, it must be between 0 and 9. Therefore, b must be 0 or 8.
Q: Why is the coefficient of c -292?
A: The coefficient of c is -292 because it is a negative multiple of 4. This means that c must be a multiple of 4. However, since c is a digit, it must be between 0 and 9. Therefore, c must be 1.
Q: How did you find the possible values of a, b, and c?
A: We found the possible values of a, b, and c by analyzing the equation. We started by substituting a = 1 into the equation and then simplified it. We then found the possible values of b and c by analyzing the resulting equation.
Q: What is the final answer?
A: The final answer is 118.
Q: Can you provide more examples of three-digit positive integers abc such that 8abc = 3cba?
A: Unfortunately, there are no other three-digit positive integers abc such that 8abc = 3cba. The equation is very specific, and the only solution is 118.
Q: How can I apply this concept to other problems?
A: This concept can be applied to other problems in number theory, such as finding all three-digit positive integers abc such that 9abc = 6cba. The key is to analyze the equation, simplify it, and find the possible values of the digits.
Conclusion
In this article, we have provided a Q&A section to answer some common questions related to the problem of determining all three-digit positive integers abc such that 8abc = 3cba. We hope that this article been helpful in understanding the concept and applying it to other problems.