3-Digit Number Puzzle: Solve The Reverse And Remainder!

Hey guys! Let's dive into a fun math problem today. We're going to crack a numerical puzzle involving a 3-digit number, its reverse, division, and remainders. Sounds intriguing, right? So, buckle up and let's get started!

Understanding the Problem

So, the core of this problem revolves around finding a 3-digit number that satisfies specific conditions when divided by its reverse. Let’s break down the given information step by step:

• We need to find a 3-digit number represented in base 10.
• When this number is divided by its reverse, the quotient is 5, and the remainder is 46.
• The difference between the tens digit and the units digit of the original number is 2.

To make things clearer, let's represent the 3-digit number as abc, where a represents the hundreds digit, b represents the tens digit, and c represents the units digit. So, the number can be written as 100a + 10b + c. The reverse of this number would be cba, which can be written as 100c + 10b + a. Now, let's translate the given information into mathematical equations.

First, the division condition tells us that:

100a + 10b + c = 5 * (100c + 10b + a) + 46

Second, the difference between the tens and units digits gives us:

b - c = 2

Now, we have two equations, and we need to find the values of a, b, and c that satisfy both. Let's simplify the first equation and see where it leads us. This problem combines number theory concepts with algebraic manipulation, making it a great exercise for problem-solving skills. Remember, the key is to break down the problem into smaller, manageable parts and then use the given information to form equations. Keep those thinking caps on, folks!

Setting up the Equations

Okay, so let's dive deeper into setting up these equations. As we mentioned earlier, representing the 3-digit number as abc helps a lot. This means our number is 100a + 10b + c, and its reverse is 100c + 10b + a. We've got two crucial pieces of information:

1. The division condition: When the original number is divided by its reverse, we get a quotient of 5 and a remainder of 46. This translates to the equation:

   100a + 10b + c = 5 * (100c + 10b + a) + 46

2. The digit difference: The difference between the tens digit (b) and the units digit (c) is 2. This gives us:

   b - c = 2

Now, let's simplify the first equation. Expanding the right side, we get:

100a + 10b + c = 500c + 50b + 5*a + 46

Now, let’s rearrange the terms to group the variables together:

95a - 40b - 499*c = 46

So, we now have two equations:

1. 95a - 40b - 499*c = 46
2. b - c = 2

We need to solve this system of equations. Notice that we have three unknowns (a, b, and c) but only two equations. This means we'll need to use some logical deduction and the properties of digits to find our solution. Remember, a, b, and c are digits, so they can only be integers from 0 to 9. Also, a cannot be 0 since it's the leading digit of a 3-digit number. Keep these constraints in mind as we move forward. We're building a solid foundation here, guys! Next up, we'll look at how to solve these equations and find the digits.

Solving the Equations

Alright, let's get our hands dirty and tackle these equations. We have the following:

1. 95a - 40b - 499*c = 46
2. b - c = 2

From the second equation, we can express b in terms of c:

b = c + 2

Now, let's substitute this into the first equation:

95a - 40(c + 2) - 499*c = 46

Expand and simplify:

95a - 40c - 80 - 499*c = 46

95a - 539c = 126

Now we have a single equation with two variables: 95a - 539c = 126. This looks a bit more manageable! Now, we need to use the fact that a and c are digits (0-9) to find possible solutions. Also, remember that a cannot be 0.

Let's rearrange the equation to isolate a:

95a = 539c + 126

a = (539*c + 126) / 95

Since a must be an integer, the expression (539*c + 126) must be divisible by 95. We can now try different values of c (from 0 to 9) and see if we get an integer value for a. Let’s create a small table to track our progress:

We see that when c = 1, we get a = 7. Now we can find b using the equation b = c + 2:

b = 1 + 2 = 3

So, we have a = 7, b = 3, and c = 1. Let's see if this works! We’re getting closer to the solution, guys. It's all about methodical problem-solving!

Verifying the Solution

Now comes the crucial part: verifying our solution! We found that a = 7, b = 3, and c = 1. So, our 3-digit number is 731. Let's check if it satisfies the given conditions:

1. The reverse of 731 is 137.
2. Divide 731 by 137: 731 ÷ 137 = 5 with a remainder of 46. This matches the first condition!
3. The difference between the tens digit and the units digit: 3 - 1 = 2. This also matches the second condition!

Since our values satisfy both conditions, we have found the solution! The 3-digit number is 731. This step is super important because it confirms that our calculations and deductions were correct. It's always a good idea to double-check your work in math problems, especially when dealing with multiple conditions. We’ve successfully navigated through the equations and conditions, guys. Give yourselves a pat on the back!

Conclusion

So, the final answer to our puzzle is 731! We successfully found the 3-digit number that, when divided by its reverse, gives a quotient of 5 and a remainder of 46, with the difference between the tens and units digits being 2. This problem was a fantastic exercise in combining algebraic manipulation with logical deduction. We started by breaking down the problem into smaller parts, setting up equations based on the given information, and then solving those equations using substitution and trial-and-error, guided by the constraints of the digits. We then verified our solution to ensure it met all the conditions.

Remember, guys, problem-solving in mathematics (and in life!) often involves breaking down complex issues into simpler steps, using the information you have to build a path forward, and always verifying your results. I hope you enjoyed this numerical journey as much as I did! Keep those brains buzzing with curiosity and keep tackling those puzzles. You’ve got this! And remember, every problem solved is a step forward in your mathematical journey. Until next time, keep puzzling!