Math Problem: Find The Number With Quotient 5 & Remainder 98

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Hey guys! Let's dive into a cool math problem today. We're going to figure out how to find a non-zero natural number that, when divided by a two-digit number, gives us a quotient of 5 and a remainder of 98. Sounds like a fun challenge, right? So, let's break it down step by step.

Understanding the Problem

Before we jump into solving, let’s make sure we really understand what the question is asking. We need to find a number (let's call it 'N') that fits this description: When N is divided by a two-digit number (let's call it 'D'), the result is a quotient of 5 and a remainder of 98. This is super important: the remainder (98) is always smaller than the divisor (D). This gives us a crucial clue to solve this puzzle!

  • Keywords to keep in mind: We're looking for a natural number, which means it's a positive whole number (1, 2, 3, and so on). It's also non-zero, meaning it's not 0. The divisor must be a two-digit number, so it falls between 10 and 99.
  • Why this matters: Understanding these terms is key because it narrows down our possibilities significantly. If we were looking for any number, it would be an impossible task! By focusing on these restrictions, we're setting ourselves up for success. So, let's keep these definitions in our back pockets as we move forward. Remember, in math, paying attention to the details of the problem is half the battle!
  • Remainder Rule: The remainder is always less than the divisor. This is the golden rule here. Our remainder is 98, so we instantly know our two-digit divisor must be greater than 98. This is a HUGE clue that simplifies our search dramatically. Can you already see where we're going with this? Let's keep digging!

Setting Up the Equation

Okay, so now that we’ve got a handle on what the question is asking, let's translate those words into a mathematical equation. This is a super powerful technique in math – turning a word problem into something we can actually solve with symbols and numbers. Think of it like decoding a secret message!

  • The Division Relationship: The core idea here is division. We know that when we divide N by D, we get a quotient and a remainder. The standard way to express this is: N = (D * Quotient) + Remainder. This formula is the backbone of our solution, so let's make sure we're comfortable with it.
  • Plugging in what we know: In our specific problem, we know the quotient is 5 and the remainder is 98. So, we can plug those values into our formula: N = (D * 5) + 98. See how we're starting to turn the problem into something more concrete? We've gone from a wordy description to a neat little equation.
  • Why this helps: This equation is awesome because it shows the direct relationship between N and D. If we can figure out what D is, we can easily calculate N. It's like having a treasure map that leads straight to the answer! By using this equation, we've taken a big step towards cracking the code. Now, let's see what we can figure out about D.

Finding the Two-Digit Number (D)

This is where the detective work really begins! We know that 'D' is a two-digit number, and we also know something super important about it because of the remainder. Remember the golden rule: the remainder (98) must be smaller than the divisor (D). This is a HUGE clue!

  • The Remainder's Clue: Because our remainder is 98, the two-digit number 'D' must be greater than 98. Think about it: you can't have a remainder larger than what you're dividing by. It's like trying to fit 100 marbles into a bag that only holds 99 – it just won't work! So, this instantly limits our possibilities for 'D'.
  • Two-Digit Restriction: We also know that 'D' has to be a two-digit number. This means it has to be less than 100. So, we've got two crucial pieces of information: D > 98 and D < 100. Can you see the solution forming?
  • Putting it together: There's only one two-digit number that's greater than 98 and less than 100. That number is 99! So, we've solved the mystery of 'D': D = 99. This is a major breakthrough. We've gone from a range of possibilities to pinpointing a single value for our divisor. Now, the rest should be smooth sailing!

Calculating the Natural Number (N)

Alright, we've found 'D'! We know the two-digit number is 99. Now we can finally calculate the natural number 'N' we've been searching for. Remember our equation from earlier? N = (D * 5) + 98. Let's plug in D = 99 and see what we get.

  • Plugging in D: Replacing 'D' with 99 in our equation gives us: N = (99 * 5) + 98. Now it's just a matter of doing the arithmetic. Time to brush up on those multiplication and addition skills!
  • Order of Operations: Remember, we need to follow the order of operations (PEMDAS/BODMAS). This means we multiply before we add. So, first, we calculate 99 * 5. This equals 495. Then, we add 98 to that result.
  • The Final Calculation: Adding 98 to 495 gives us 593. So, N = 593! We've found our number! This is the natural number that, when divided by 99, gives a quotient of 5 and a remainder of 98. Hooray!

Verifying the Solution

It's always a good idea to double-check your work, especially in math! Let's make sure that 593 really does fit the conditions of the problem. We need to verify that when we divide 593 by 99, we get a quotient of 5 and a remainder of 98.

  • Performing the Division: Let's do the division: 593 ÷ 99. You can do this with long division, a calculator, or even a little mental math. The important thing is to make sure we get the right quotient and remainder.
  • Checking the Result: 593 divided by 99 gives us 5 with a remainder of 98. This is exactly what we were looking for! So, our solution, N = 593, is correct. High five!
  • Why verify?: Verifying your solution is a crucial step in problem-solving. It helps catch any small errors you might have made along the way. Think of it as the final polish on your masterpiece. It ensures that you're not just getting an answer, but you're getting the right answer.

Conclusion

Awesome job, guys! We successfully found the natural number 593 that, when divided by 99, gives a quotient of 5 and a remainder of 98. We did it by breaking down the problem, setting up an equation, using the clues from the remainder, and carefully calculating the result. And, of course, we verified our answer to make sure we nailed it!

Remember, the key to solving these kinds of problems is to read carefully, understand the relationships between the numbers, and translate the words into math. Keep practicing, and you'll become a math whiz in no time! If you want more problems like this, let me know, and we can tackle some more together. Happy problem-solving!