Finding The Biggest Number: Math Challenge!

Hey guys! Let's dive into a fun little math puzzle. We're on a mission to find the largest three-digit number. But, there's a catch! This number needs to have distinct digits (meaning no repeating digits) and when we divide it by 12, it leaves a remainder of 5. Sound like a challenge? Absolutely! But don't worry, we'll break it down step-by-step, making it super easy to understand. This is the type of problem that can seem a little daunting at first glance, but with a systematic approach, we can totally crack it. Think of it like a treasure hunt – we have clues (the rules of the problem) and we need to follow them to find the hidden treasure (our answer!). So, grab your thinking caps, and let's get started! We are going to go through this together, from the basics to the more complex concepts involved, so you will be able to solve similar problems. This particular problem is a great example of how even seemingly complex mathematical concepts can be broken down into manageable parts. It encourages logical thinking and problem-solving skills, which are super valuable, not just in math, but in life in general.

Before we start, it's important to understand what a remainder is. When you divide one number by another, the remainder is the 'leftover' part that doesn't divide evenly. For example, if you divide 17 by 5, you get 3 with a remainder of 2. We will use this concept a lot in our journey to find the right answer. Our goal is to find the largest possible three-digit number that follows these rules. We need to keep in mind that we need different digits, so no repeats. This adds an extra layer of thinking, but it makes the puzzle even more interesting! It's like building with Lego bricks – you can only use each brick once. The remainder of 5 when divided by 12 gives us a really important piece of information, and it is crucial to solving the problem. So, let's gear up and get into the core of the solution!

Breaking Down the Problem: The Strategy

Alright, let's get our game plan in motion. The key to solving this type of problem is to work backwards. We want the largest three-digit number, so let's start by considering the largest possible digits. We'll start with the biggest possible hundreds digit, then the biggest possible tens digit, and finally the biggest possible units digit, all while making sure we satisfy our rules. This is a very common strategy used in solving math problems, particularly those related to number theory. It's about being systematic and methodical – thinking through each step carefully. We also need to remember that the number has to have a remainder of 5 when divided by 12. So, let's think about what that means. If a number leaves a remainder of 5 when divided by 12, we can express that number as 12*k + 5, where 'k' is any whole number. This is a crucial concept. Every number that satisfies our conditions can be described this way. In essence, we're finding numbers that are 5 more than a multiple of 12. So, if we take any multiple of 12 (12, 24, 36, etc.) and add 5 to it, we get a number that fits our criteria. This is how we'll narrow down our choices. Now, we'll use trial and error with an intelligent strategy. We are going to try the largest possible three-digit numbers with different digits, checking if the remainder is 5 when divided by 12. This is where the real work begins. It's like testing out different keys until we find the one that fits the lock. The process involves a bit of calculation, a bit of patience, and a good understanding of remainders. We will start from the top and go down until we find our answer! Let's get our hands dirty, guys!

Starting with the Highest Digits

So, let's start by assuming the hundreds digit is 9 (the biggest digit). Now, to get the largest possible number, we'll try to put the next largest digits in the tens and units place. We'll start with 8 in the tens place. So far, we're considering numbers like 98x. If we add 5 to a multiple of 12, the last digit will often vary, but we need to check the numbers which have distinct digits. The next digit we would try would be 7, so we are now considering numbers like 987. Can we find any numbers fitting our criteria? We would start checking numbers starting from 980, 981, 982 and so on, until we reach 987. When we start checking with 980 and add 5, then we get 985, which has a remainder of 1 when divided by 12. So, 985 does not satisfy our condition, but we're getting closer. We continue the same process.

Let's continue testing for the next largest possible digits after the hundreds place is 9 and the tens place is 8. We already know 985 does not fit our criteria. Let's try the next possible digit, which is 6. So the number is 986. Since we want the remainder to be 5 when divided by 12, we could write it as 986 divided by 12 = 82 with a remainder of 2. So, 986 does not satisfy our conditions. Next, the units digit, could be 4. But 984 divided by 12 gives 82 and the remainder is 0. This shows that 989 is also not the correct answer. Keep in mind, the main rule we are looking for is distinct digits with the remainder being 5 when divided by 12.

Refining Our Search

Now, let's adjust our approach a bit. Instead of starting with 98x, we should consider what the last digit must be to have a remainder of 5 when divided by 12. We can use the formula 12*k + 5. But we need to make sure the digits are distinct. So, we'll test some numbers. Remember that the number must be less than 999. If we try 995, that is not correct, as it has repeating numbers. Let us try 989, which also doesn't satisfy the criteria. But we're getting closer! This is all part of the process of finding the solution. Sometimes it involves trial and error, but always in a systematic way. When it comes to complex problems, this strategy is your best friend. We will look for the next possible numbers starting from the largest possible number. We are basically doing backwards engineering. Think of a detective solving a case – they gather clues (the rules), eliminate suspects (numbers that don't fit), and eventually pinpoint the culprit (our answer). And remember, patience is key! Sometimes, the answer isn't obvious, and we might need to explore several possibilities before we find it. We will need to be thorough and methodical in our calculations, and we will get to the answer.

So, let's go back to where we started. We know that our number must be in the form of 12*k + 5. This is very important to remember. We are looking for the largest three-digit number. We'll go backwards and start from the largest possible three-digit number and go down to smaller numbers. So let's assume the hundreds digit is 9. The largest digit possible for the tens digit is 8. Therefore, we are checking 98x. Since the remainder must be 5 when divided by 12, 98x = 12 * k + 5. After we divide 989 by 12 we get a remainder of 5. So, we have our answer! We have found it!

The Answer Revealed!

After all of this, we now have our answer! The largest three-digit number with distinct digits that leaves a remainder of 5 when divided by 12 is 989!

Congratulations, guys! We've solved it! We used a combination of logical thinking, strategic testing, and a good understanding of remainders to crack the code. You've shown that with a systematic approach, any math problem can be solved. Math can be fun when approached step-by-step like we just did. Keep practicing, keep exploring, and you'll become math wizards in no time! Keep in mind that you can apply this method to other problems with different conditions. This is a core mathematical concept and the knowledge is valuable in real life too! Keep up the great work and happy problem-solving!