Find The Divisor: Remainders 32, 42, 52

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Find the Divisor: Remainders 32, 42, 52

Hey guys! Let's dive into this interesting math problem where we need to find a natural number that divides three given numbers (758, 647, and 899) and leaves specific remainders (32, 42, and 52). This might sound a bit tricky at first, but don't worry, we'll break it down step by step so it's super easy to understand. This is the type of problem that combines a bit of arithmetic with some logical thinking, making it a fantastic exercise for your brain. So, grab your thinking caps, and let’s get started!

Understanding the Problem

The core of this problem revolves around the concept of remainders in division. When a number (the dividend) is divided by another number (the divisor), the remainder is what's left over if the division isn't exact. In our case, we have three divisions happening:

  • 758 divided by our mystery number leaves a remainder of 32.
  • 647 divided by the same number leaves a remainder of 42.
  • 899 divided by that same number leaves a remainder of 52.

Our mission, should we choose to accept it (and we do!), is to find that mystery divisor. To tackle this, we'll use a clever trick: focusing on what happens before the remainder. Essentially, we're going to subtract the remainders from the original numbers. This gives us numbers that are perfectly divisible by our mystery divisor. This is the key to unlocking our problem, so make sure you've got this concept down! It's like finding puzzle pieces that fit perfectly together.

Step 1: Subtracting the Remainders

Okay, the first thing we need to do is get rid of those pesky remainders. This will help us find a common ground for our three numbers. We're going to subtract each remainder from its corresponding original number:

  • 758 - 32 = 726
  • 647 - 42 = 605
  • 899 - 52 = 847

What we've done here is create three new numbers (726, 605, and 847) that are each perfectly divisible by the natural number we're trying to find. Think of it like this: if dividing 758 leaves a remainder of 32, then the rest of the number (726) must be cleanly divisible. The same logic applies to the other numbers. Now, we've narrowed down our search – instead of dealing with remainders, we're looking for a common divisor of these three new numbers. This makes our problem much more manageable.

Step 2: Finding the Greatest Common Divisor (GCD)

Now that we have three numbers (726, 605, and 847) that are perfectly divisible by our mystery divisor, our next step is to find the greatest common divisor (GCD) of these numbers. The GCD is the largest number that divides evenly into all three. It’s like finding the biggest building block that can be used to construct all three numbers. There are a couple of ways we can find the GCD, but a common method is using the Euclidean algorithm. This method involves repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until we reach a remainder of 0. The last non-zero remainder is our GCD.

Let's start by finding the GCD of 726 and 605 using the Euclidean algorithm:

  1. 726 ÷ 605 = 1 remainder 121
  2. 605 ÷ 121 = 5 remainder 0

So, the GCD of 726 and 605 is 121. Now, we need to find the GCD of this result (121) and our third number (847):

  1. 847 ÷ 121 = 7 remainder 0

Aha! The GCD of 121 and 847 is 121. This means the greatest common divisor of 726, 605, and 847 is 121. We're getting closer to our solution!

Step 3: Verifying the Solution

We've found a potential answer: 121. But before we celebrate, we need to make sure it actually works! Remember, the original problem stated that our mystery number leaves specific remainders when dividing 758, 647, and 899. So, let's put 121 to the test. We need to check if the remainders match the ones given in the problem:

  • 758 ÷ 121 = 6 remainder 32 (This matches!)
  • 647 ÷ 121 = 5 remainder 42 (This matches too!)
  • 899 ÷ 121 = 7 remainder 52 (And this one matches as well!)

Fantastic! Our calculations are correct. Dividing each of the original numbers by 121 gives us the remainders specified in the problem. This confirms that 121 is indeed the natural number we were looking for. It's always a good idea to verify your solution, especially in math problems. This gives you confidence in your answer and helps you avoid making silly mistakes. Think of it as the final checkmark on your puzzle, ensuring all the pieces fit perfectly.

Step 4: Considering Other Possible Solutions

While 121 works perfectly, there's one more thing we need to think about. Could there be other possible solutions? Remember, the remainders (32, 42, and 52) must be smaller than the divisor. This is a crucial point! If the remainder were larger than the divisor, we could divide further.

Let's look at our remainders: 32, 42, and 52. The largest of these is 52. This means our divisor must be greater than 52. Our GCD, 121, is certainly greater than 52, which is good. But, are there any other divisors of 726, 605, and 847 that are also greater than 52? To find out, we could look at the factors of 121. The factors of 121 are 1 and 11 and 121.

Let's check them:

  • 121: We already know this works.
  • 11: is less than all the remainders.
  • 1: is less than all the remainders.

Therefore, 121 is the only natural number that satisfies the conditions of the problem. This step is important because it shows we've thought critically about the problem and haven't just stopped at the first solution we found. We've made sure our answer is not only correct but also the only correct answer.

Conclusion

So, there you have it! We've successfully navigated through this number theory problem and found the natural number that divides 758, 647, and 899, leaving remainders of 32, 42, and 52 respectively. The answer, as we've confirmed, is 121.

We did it by:

  1. Subtracting the remainders to get numbers perfectly divisible by our target.
  2. Finding the Greatest Common Divisor (GCD) of those new numbers.
  3. Verifying that our GCD actually produced the correct remainders.
  4. Considering other possible solutions to ensure we had the only answer.

This type of problem is a great exercise in logical thinking and applying math concepts in a creative way. Remember, the key is to break down the problem into smaller, manageable steps. And don't be afraid to think outside the box – sometimes the most elegant solutions come from a slightly different perspective. Keep practicing, keep exploring, and most importantly, keep having fun with math! You guys nailed it!