Maximum Number Of Books Nazlı Has A Math Problem Solution

by Scholario Team 58 views

Hey guys! Today, we're diving into a cool math problem that involves figuring out the maximum number of books Nazlı has. This is a fun one because it mixes basic division concepts with a bit of logical thinking. So, let's break it down step by step and get to the solution together!

Understanding the Problem

Okay, so here’s the deal. Nazlı is grouping her books. When she tries to group them in sets of four or seven, she always has three books left over. We also know that the total number of books is a two-digit number. Our mission is to find the largest possible number of books Nazlı could have.

Breaking Down the Information

Let's highlight the important stuff:

  • Grouping by Four: When Nazlı groups her books by four, there are three left over.
  • Grouping by Seven: When she groups them by seven, there are also three left over.
  • Two-Digit Number: The total number of books is a two-digit number (meaning it’s between 10 and 99).
  • Maximum Number: We need to find the highest possible number of books that fits these conditions.

Why This Problem Matters

Problems like these aren't just about math; they're about problem-solving skills. They help us think logically and break down complex information into manageable parts. Plus, they're kinda fun once you get the hang of them!

Finding the Solution

Alright, let’s get our hands dirty and solve this thing. We need to figure out a number that, when divided by both 4 and 7, leaves a remainder of 3. This involves a bit of number theory, but don't worry, we'll make it super clear.

Step 1 The Least Common Multiple (LCM)

First things first, we need to find the Least Common Multiple (LCM) of 4 and 7. The LCM is the smallest number that both 4 and 7 divide into evenly. This is crucial because it gives us a baseline to work with.

  • Multiples of 4: 4, 8, 12, 16, 20, 24, 28, ...
  • Multiples of 7: 7, 14, 21, 28, 35, ...

Notice that 28 is the smallest number that appears in both lists. So, the LCM of 4 and 7 is 28. This means any multiple of 28 will be divisible by both 4 and 7.

Why is LCM Important?

The LCM helps us find numbers that are divisible by both 4 and 7. Since we need a remainder of 3, we’ll use the LCM as a foundation and then adjust for the remainder.

Step 2 Adjusting for the Remainder

Now that we have the LCM (28), we need to account for the remainder of 3. This is the tricky part, but stick with me.

We know that when Nazlı groups her books, there are 3 left over. So, we need to find a number that is a multiple of 28, plus 3. Let's try a few multiples of 28 and add 3 to each:

  • 28 * 1 + 3 = 31
  • 28 * 2 + 3 = 59
  • 28 * 3 + 3 = 87
  • 28 * 4 + 3 = 115 (This is a three-digit number, so it doesn't fit our condition)

So, we have potential candidates: 31, 59, and 87. These numbers all leave a remainder of 3 when divided by both 4 and 7.

Step 3 Finding the Maximum Two-Digit Number

Remember, we need to find the maximum number of books Nazlı could have, and this number must be a two-digit number. From our list (31, 59, 87), the largest two-digit number is 87.

Therefore, the maximum number of books Nazlı has is 87.

Checking Our Work

It’s always a good idea to double-check our answer. Let's see if 87 fits our conditions:

  • 87 ÷ 4 = 21 remainder 3 (Correct!)
  • 87 ÷ 7 = 12 remainder 3 (Correct!)

Yep, 87 checks out. We’ve nailed it!

Alternative Approach Listing and Checking

If the LCM method feels a bit complicated, here’s another way to think about it: List numbers that leave a remainder of 3 when divided by 4 and 7, and see which one is the largest two-digit number.

Listing Numbers with Remainder 3 When Divided by 4

Let’s start by listing numbers that leave a remainder of 3 when divided by 4:

  • 4 * 1 + 3 = 7
  • 4 * 2 + 3 = 11
  • 4 * 3 + 3 = 15
  • ...

We can continue this pattern to find more numbers.

Listing Numbers with Remainder 3 When Divided by 7

Now, let's list numbers that leave a remainder of 3 when divided by 7:

  • 7 * 1 + 3 = 10
  • 7 * 2 + 3 = 17
  • 7 * 3 + 3 = 24
  • ...

Combining the Lists and Finding the Maximum

Now, we need to find a number that appears in both lists and is the largest two-digit number. This might take a bit of manual checking, but it’s a solid method.

If we extend the lists, we’ll find that 87 appears in both and is the highest two-digit number that fits:

  • Numbers with remainder 3 when divided by 4: ..., 75, 79, 83, 87, 91, 95, 99
  • Numbers with remainder 3 when divided by 7: ..., 73, 80, 87, 94

Again, we see that 87 is the answer!

Common Mistakes to Avoid

When tackling problems like this, there are a few common pitfalls to watch out for. Let's make sure we sidestep them:

Misunderstanding the Remainder

One common mistake is not fully grasping the concept of remainders. Remember, the remainder is what's left over after division. So, having a remainder of 3 means the number is 3 more than a multiple of both 4 and 7.

Forgetting the Two-Digit Constraint

It’s easy to get caught up in the math and forget that we’re looking for a two-digit number. Always keep the problem's conditions in mind to avoid incorrect answers. For instance, we calculated 115 as a potential answer, but it’s a three-digit number, so it doesn’t fit.

Not Checking the Answer

Always, always, always check your answer! It takes just a minute to verify that 87 divided by 4 and 7 leaves a remainder of 3. This simple step can save you from making a careless mistake.

Real-World Applications

You might be thinking, "Okay, this is cool, but when will I ever use this in real life?" Well, problem-solving skills are valuable everywhere! But here are a few scenarios where this kind of thinking can come in handy:

Organizing Items

Imagine you’re organizing items into boxes. If you need to distribute items evenly and have some leftovers, this concept applies directly.

Scheduling Tasks

When scheduling tasks that need to be repeated at different intervals, understanding remainders can help you sync up those tasks efficiently.

Computer Science

In computer science, the concept of remainders is used in many algorithms, such as those for data encryption and error detection.

Everyday Math

Even splitting bills with friends or figuring out how many packs of something you need for a project can involve similar logic.

Conclusion

So, guys, we’ve successfully tackled a tricky math problem and found that Nazlı has a maximum of 87 books. We walked through the process of finding the Least Common Multiple, adjusting for the remainder, and checking our answer. Remember, problem-solving is a skill that gets better with practice. Keep challenging yourself with these kinds of questions, and you’ll become a math whiz in no time!

If you enjoyed this breakdown, give it a thumbs up and share it with your friends. And if you have any other math problems you’d like us to solve, drop them in the comments below. Happy problem-solving!