Find Pedro's Address: A Four-Digit Math Puzzle

Hey guys! Ever been in a situation where you need to send a letter to a friend but can't quite remember their full address? Our friend Juan is facing that exact problem! He needs to mail a letter to Pedro, who lives on Reforma Street. Juan knows Pedro's house number has four digits, is a multiple of both 5 and 7, and ends in 0. Let's put on our detective hats and use a little math to help Juan figure out Pedro's house number!

Cracking the Code: Finding the House Number

Okay, let's break down the clues we have about Pedro's house number. This is where the fun begins! We know the house number is a four-digit number, meaning it falls somewhere between 1000 and 9999. This gives us a starting range to work with. But the real game-changers are the divisibility rules. Since the number is a multiple of 5, we instantly know it must end in either 0 or 5. But wait! We also know the last digit is 0, so that narrows it down even further. This is a crucial piece of information because it simplifies our search significantly. Think of it like unlocking the first level in a video game – we're already making progress!

The next important clue is that the house number is also a multiple of 7. This is where things get a little trickier, but don't worry, we can handle it! There isn't a simple divisibility rule for 7 that's as straightforward as the one for 5. Instead, we need to use a bit of logic and maybe some trial and error (but not too much, we promise!). We're looking for a number that is divisible by both 5 and 7, which means it must also be divisible by their least common multiple (LCM). Finding the LCM will give us a crucial stepping stone in our quest.

To find the LCM of 5 and 7, we can simply multiply them together since they are both prime numbers. So, 5 multiplied by 7 equals 35. This tells us that Pedro's house number must be a multiple of 35. Now we're getting somewhere! But remember, it also has to be a four-digit number ending in 0. This means we need to find a multiple of 35 that fits these criteria. It’s like we are solving a puzzle within a puzzle, which makes it even more exciting!

Now, let’s combine all the clues. We need a four-digit number, a multiple of 35, and it must end in 0. To make things even easier, let's think about multiples of 10, since any number ending in 0 is a multiple of 10. So, we are essentially looking for a multiple of both 35 and 10. The LCM of 35 and 10 will give us the key to unlock this final piece of the puzzle. To find the LCM of 35 and 10, we can list their multiples and find the smallest one they have in common or use the prime factorization method. Either way, this step will lead us closer to the answer.

Narrowing Down the Possibilities: Multiples and Logic

Let's dive deeper into narrowing down the possibilities. We know Pedro's house number is a multiple of 35 and ends in 0. This means it's also a multiple of 10. Finding the Least Common Multiple (LCM) of 35 and 10 is key. The multiples of 35 are 35, 70, 105, and so on. The multiples of 10 are 10, 20, 30, and so on. The smallest number that appears in both lists is 70. So, the LCM of 35 and 10 is 70. This tells us that Pedro's house number must be a multiple of 70.

Now we need to find a four-digit multiple of 70. Think about it this way: we're essentially climbing a staircase, where each step is 70 units high. We need to find a step that lands us within the four-digit range (1000-9999). To do this, we can divide the smallest four-digit number (1000) by 70. 1000 divided by 70 is approximately 14.28. Since we can't have a fraction of a step, we round up to the nearest whole number, which is 15. This means the 15th multiple of 70 might be our starting point. Let's calculate 15 times 70. 15 multiplied by 70 is 1050. Aha! We have our first four-digit multiple of 70.

But is this Pedro's house number? We need to keep exploring. We've found the smallest four-digit multiple of 70, but there could be others. Think of it like searching for buried treasure – we've found one possible spot, but we need to make sure there isn't a better one nearby. So, we continue climbing the staircase, taking steps of 70, until we reach the end of our four-digit range. We want to find all the multiples of 70 that fall between 1000 and 9999. This requires a bit more calculation, but we're on the right track! Finding all these multiples gives us a set of candidates, and then we can check if any other conditions apply to narrow it down even further.

We can continue to add 70 to our previous multiple (1050) to find the next one. 1050 plus 70 is 1120. And so on. We keep adding 70 until we get a number greater than 9999. To make this process a bit faster, we can also divide the largest four-digit number (9999) by 70. 9999 divided by 70 is approximately 142.84. So, we round down to 142, since we only want whole multiples. This means the 142nd multiple of 70 is the largest four-digit multiple. Now we know that all the multiples of 70 from the 15th (1050) to the 142nd will be potential candidates for Pedro's house number. This range gives us a manageable set of numbers to investigate.

The Final Deduction: Finding the Exact Number

Alright, guys, we've got a solid list of potential house numbers – all the multiples of 70 between 1050 (15 * 70) and 9940 (142 * 70). That’s a great achievement, but let’s not celebrate just yet! We need to make the final deduction to pinpoint Pedro's exact address. It’s like being in the last scene of a mystery movie, where all the clues come together. We've laid the groundwork; now, the moment of truth!

To crack this final part, we need to revisit the original problem statement. Juan knew Pedro's house number had four digits, was a multiple of 5 and 7 (which we've already used to determine it’s a multiple of 70), and importantly, the last digit is 0. This last bit is something we’ve already factored in, but it’s a good reminder that we’re on the right path. However, let's imagine there was another clue we had overlooked. Perhaps Juan also mentioned that the sum of the digits in Pedro's house number is a certain value, or that the number is greater than a specific number. These additional pieces of information would act like a magnifying glass, helping us zoom in on the correct answer.

For example, suppose Juan remembered that the sum of the digits in Pedro's house number is 6. Now, we have a new criterion to check against our list of multiples of 70. We would need to add the digits of each potential house number and see if the sum equals 6. This might sound tedious, but it’s a systematic way to eliminate possibilities. It's like using a fine-toothed comb to sift through our options.

Let's go back to our list of multiples of 70. We have numbers like 1050, 1120, 1190, and so on. To apply our hypothetical