Finding The Divisor: A Division With Remainder Problem

by Dimemap Team 55 views

Hey guys! Let's dive into a cool math problem today where we'll figure out how to find the divisor in a division problem that has a remainder. We've got a dividend of 427, a remainder of 75, and a mystery divisor that's a two-digit number with both digits the same. Sounds like fun, right? Let's break it down and solve it together!

Understanding the Problem

Okay, so our main goal here is to find the divisor. To get started, let's really understand what we're dealing with. We know a few things for sure:

  • The dividend is 427. This is the number we're dividing.
  • The remainder is 75. This is what's left over after the division.
  • The divisor is a two-digit number, and both digits are the same (like 11, 22, 33, and so on).

Knowing these pieces of information is crucial. The relationship between the dividend, divisor, quotient, and remainder is key to cracking this problem. Remember the basic formula:

Dividend = (Divisor × Quotient) + Remainder

This formula is the backbone of division problems with remainders, and we'll use it to guide our thinking. Think of it like this: if you multiply the divisor by the quotient (the result of the division) and then add the remainder, you should get back the original dividend. This understanding is fundamental to our approach.

Now, let's consider the remainder. The remainder is always smaller than the divisor. This is a critical piece of information! If the remainder were larger than or equal to the divisor, we could divide further. In our case, the remainder is 75, which means our divisor must be larger than 75. This significantly narrows down the possibilities for our two-digit divisor with identical digits. We can immediately eliminate 11, 22, 33, 44, 55, 66 and 77 as potential divisors. This leaves us with 88 and 99 as the only viable candidates. This simple deduction saves us a lot of time and effort!

So, with a solid grasp of the problem and a key deduction about the possible divisors, we're well-prepared to move on to the next step: using this knowledge to actually find the correct divisor. Let's keep going!

Narrowing Down the Possibilities

Alright, based on our understanding, we've figured out that the divisor has to be a two-digit number with identical digits and it must be greater than 75. This really narrows things down for us! We can quickly list the possibilities:

  • 88
  • 99

That's it! We've gone from potentially any two-digit number to just two options. This is a huge step forward. Now, how do we figure out which one is the correct divisor? This is where we'll use our understanding of the relationship between the dividend, divisor, quotient, and remainder, and a little bit of trial and error.

We know that Dividend = (Divisor × Quotient) + Remainder. We can rearrange this formula to help us test our possibilities:

Dividend - Remainder = Divisor × Quotient

This rearranged formula tells us that if we subtract the remainder from the dividend, the result should be perfectly divisible by the divisor. In other words, the result should be a multiple of the divisor. This gives us a direct way to test whether 88 or 99 is the correct divisor. We'll perform the subtraction on the left side and then check if the result is divisible by each potential divisor.

Let's start by subtracting the remainder (75) from the dividend (427):

427 - 75 = 352

So, 352 must be divisible by our correct divisor. Now, we'll check if 352 is divisible by 88 and 99. This involves trying out the division and seeing if we get a whole number as the quotient. If we don't get a whole number, then that number isn't the correct divisor. If we do get a whole number, we've found our divisor! This is a straightforward process that will lead us to the solution.

By systematically narrowing down the possibilities based on the information given in the problem, we're making the problem much more manageable. Now, let's perform the divisions and see which divisor fits!

Testing the Potential Divisors

Okay, we've narrowed it down to two potential divisors: 88 and 99. We also know that 352 (which is 427 - 75) must be perfectly divisible by the correct divisor. Let's put these to the test, guys!

First, let's try dividing 352 by 88:

352 ÷ 88 = 4

Wow! It divides perfectly. We get a whole number (4) as the quotient. This is a really good sign! It suggests that 88 might indeed be our divisor. But, just to be absolutely sure, let's also test the other possibility, 99, to see if it also works. This will confirm whether we have a unique solution or if there's another possibility we need to consider.

Now, let's try dividing 352 by 99:

352 ÷ 99 = 3.555...

Uh oh! We don't get a whole number here. We get a decimal, which means 352 is not perfectly divisible by 99. This confirms that 99 cannot be the divisor in this case. It's really important to test all possibilities, even when one looks promising, just to ensure we're arriving at the correct answer and not overlooking anything.

Since 352 is divisible by 88 but not by 99, we can confidently conclude that 88 is the correct divisor. This step-by-step process of testing each potential solution is a powerful technique in problem-solving. It allows us to systematically eliminate incorrect options and zero in on the correct answer. We've now identified our divisor, but let's just do one final check to make sure everything fits perfectly within our original problem.

With 88 identified as the divisor, let's proceed to the final confirmation step to ensure the solution is correct.

Final Confirmation

Alright, we've figured out that the divisor is likely 88. But, let's do a final check to make sure everything fits together perfectly. This is super important to avoid any silly mistakes. We'll use our original formula:

Dividend = (Divisor × Quotient) + Remainder

We know:

  • Dividend = 427
  • Divisor = 88
  • Remainder = 75

We also calculated that the quotient when we divided 352 by 88 was 4. So, let's plug these values into the formula and see if it works out:

427 = (88 × 4) + 75

Now, let's do the math:

427 = 352 + 75

427 = 427

Yes! It checks out perfectly. The equation holds true. This gives us total confidence that we've found the correct divisor. The final check is a critical step because it catches any potential errors in our calculations or reasoning. It ensures that our answer is not only a possibility but the definitive solution to the problem.

By going through this confirmation process, we demonstrate a thorough understanding of the problem and the solution. It's not just about getting an answer; it's about knowing that the answer is correct. This practice is invaluable in mathematics and in problem-solving in general.

So, after this thorough verification, we can confidently state the solution:

Solution

The divisor is 88.

We did it, guys! We successfully found the divisor in this division problem with a remainder. We started by understanding the problem, narrowed down the possibilities, tested each potential solution, and then did a final check to confirm our answer. This step-by-step approach is super effective for tackling tricky math problems.

Remember, the key was to use the relationship between the dividend, divisor, quotient, and remainder, and to systematically eliminate incorrect options. By breaking down the problem into smaller, manageable steps, we made it much easier to solve. Don't be afraid of complex problems! With the right approach and a bit of patience, you can conquer them. And always, always do that final check!

So, keep practicing, keep exploring, and keep those math skills sharp. You've got this!