Find 'ab' Natural Numbers: Solve 3a + 5b = 17

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Hey guys! Let's dive into a cool math problem today: finding natural numbers in the form of 'ab' when given the equation 3a + 5b = 17. This might sound a bit tricky at first, but don't worry, we'll break it down step by step. We'll explore the fundamentals of number theory, emphasizing Diophantine equations and digit representation. Let’s get started and make math a little less mysterious and a lot more fun!

Understanding the Problem

Before we jump into solving, let's make sure we really get what the question is asking. We're looking for two-digit natural numbers, which we can represent as 'ab'. Here, 'a' is the tens digit and 'b' is the units digit. These digits are natural numbers, meaning they can be 0, 1, 2, and so on. However, since 'a' is the tens digit, it can't be 0 (otherwise, it wouldn't be a two-digit number). So, 'a' can be any number from 1 to 9, and 'b' can be any number from 0 to 9. The core of this problem lies in understanding the equation 3a + 5b = 17. This equation connects our two digits, 'a' and 'b', and we need to find the pairs of 'a' and 'b' that make this equation true. This type of problem often involves some trial and error, but we can use logic and deduction to make the process more efficient. By the end of this explanation, you'll be comfortable with digit analysis, equation solving, and the critical skill of logical deduction in number theory.

Breaking Down the Equation

Okay, so we have the equation 3a + 5b = 17. To solve this, we need to think about the possible values for 'a' and 'b'. Remember, 'a' and 'b' are digits, meaning they are whole numbers between 0 and 9. The trick here is to realize that we can't just plug in any numbers – they have to fit the equation perfectly. A key strategy in tackling these types of problems is to isolate one of the variables and express it in terms of the other. For instance, we could rearrange the equation to solve for 'a' or 'b'. Let’s solve for 'a' first:

3a = 17 - 5b a = (17 - 5b) / 3

Now, we have 'a' in terms of 'b'. This is super helpful because it tells us that the value of 'a' depends on the value of 'b'. The next step involves using modular arithmetic implicitly, where we consider remainders upon division. This is a crucial technique in solving Diophantine equations, which are polynomial equations where only integer solutions are of interest. The ability to isolate variables and understand their relationships is a cornerstone of algebraic problem-solving.

Finding Possible Values for 'b'

Now that we have a = (17 - 5b) / 3, let's think about what 'b' can be. Since 'a' has to be a whole number (because it's a digit), that means (17 - 5b) must be perfectly divisible by 3. This is a crucial point! We can’t have any remainders. So, we’ll test values for 'b' (remember, b can be 0, 1, 2, ..., 9) and see which ones make (17 - 5b) a multiple of 3. This process uses the mathematical concept of divisibility rules, which are shortcuts for determining if one number is divisible by another. By applying these rules, we can efficiently narrow down the possible values of 'b'. This step highlights the importance of systematic testing in mathematical problem-solving, especially when dealing with a limited set of possible solutions.

Let’s try some values for 'b':

  • If b = 0, then 17 - 5(0) = 17, which is not divisible by 3.
  • If b = 1, then 17 - 5(1) = 12, which is divisible by 3! This looks promising.
  • If b = 2, then 17 - 5(2) = 7, which is not divisible by 3.
  • If b = 3, then 17 - 5(3) = 2, which is not divisible by 3.

It seems like b = 1 is the only value that works within the first few tries. But let's continue to make sure we haven't missed anything. We're employing a trial-and-error strategy, but it’s a smart trial-and-error, guided by our divisibility rule. This is a common approach in number theory, where direct algebraic manipulation might be complex, and testing potential candidates provides a more straightforward solution path.

Calculating 'a' and Checking the Solution

Alright, we found that b = 1 is a possible solution. Now, let's plug that back into our equation for 'a':

a = (17 - 5b) / 3 a = (17 - 5(1)) / 3 a = (17 - 5) / 3 a = 12 / 3 a = 4

So, we have a = 4 and b = 1. This means our two-digit number 'ab' is 41. But, we need to verify our solution to ensure that it actually satisfies the original equation. This is a crucial step in problem-solving – always double-check your work!

Let's plug a = 4 and b = 1 back into the original equation, 3a + 5b = 17:

3(4) + 5(1) = 12 + 5 = 17

It works! So, 41 is indeed a solution. The process of solution verification is essential to avoid errors and ensure accuracy, particularly in competitive mathematics and real-world applications. This step reinforces the importance of rigorous thinking and attention to detail.

Are There Any Other Solutions?

Now, the big question: is 41 the only solution? To be absolutely sure, we need to think about whether there could be any other values of 'b' that would make (17 - 5b) divisible by 3 and result in a whole number for 'a'. Remember, 'b' can only be digits from 0 to 9. Let’s go through the remaining values systematically.

We’ve already tested b = 0, 1, 2, and 3. Let’s continue:

  • If b = 4, then 17 - 5(4) = -3, which is divisible by 3, but it gives us a = -1. This doesn’t work because 'a' must be a positive digit.
  • If we try any larger values for 'b', such as 5, 6, 7, 8, or 9, the result (17 - 5b) will be negative and lead to a negative value for 'a', which isn't possible.

This systematic approach demonstrates the use of exhaustive search, a method where every possible solution is checked to ensure that no solutions are missed. While exhaustive search can be time-consuming for larger problems, it is perfectly manageable for a small set of possibilities. This method illustrates the principle of completeness in mathematical problem-solving, where the goal is to find not just a solution, but all solutions.

The Final Answer

After carefully checking all the possibilities, we've found that there's only one solution that fits our conditions. The natural number in the form of 'ab' that satisfies the equation 3a + 5b = 17 is 41. That’s it! We’ve successfully navigated this problem by breaking it down into smaller, manageable steps, using a mix of algebraic manipulation, divisibility rules, and good old-fashioned logic. Remember, the key to solving problems like these isn't just about knowing the formulas, it's about thinking critically and systematically. The journey to the final answer showcases several critical mathematical skills, including algebraic manipulation, divisibility testing, systematic problem-solving, and solution verification. Mastering these skills not only helps in solving mathematical problems but also enhances logical reasoning and analytical thinking in general.

Key Takeaways

  • Understanding the Problem: Always start by making sure you fully understand what the problem is asking.
  • Breaking It Down: Complex problems become easier when you break them down into smaller steps.
  • Using Equations: Algebraic equations are powerful tools for representing relationships between numbers.
  • Trial and Error (Smartly): Don't be afraid to try different possibilities, but do it in a way that's guided by logic and rules.
  • Checking Your Work: Always, always double-check your answers to make sure they make sense.

By applying these principles, you'll be well-equipped to tackle a wide range of mathematical challenges. So, keep practicing, keep thinking, and most importantly, keep having fun with math!

I hope this explanation helped you guys understand how to solve this type of problem. Math can be challenging, but with the right approach, you can conquer anything! Keep practicing, and you'll become a math whiz in no time. If you have any more questions or want to tackle another problem, just let me know!