Finding X: 5 Divides (15 + X) Where X < 10

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Hey guys! Today, we're diving into a fun math problem where we need to find the values of 'x' that satisfy a specific condition. This condition involves divisibility, which is a crucial concept in number theory. We'll be working with natural numbers and figuring out which ones, when added to 15, result in a number that's perfectly divisible by 5. So, let's get started and break down this problem step by step!

Understanding the Problem

Let's first understand the core of the problem. We are looking for natural numbers, represented by 'x', that meet two criteria. First, 'x' must be less than 10. Second, when we add 'x' to 15, the resulting sum must be divisible by 5. In mathematical terms, this means that (15 + x) should be a multiple of 5. To solve this, we'll explore the multiples of 5 and see which values of 'x' fit our conditions. This involves a bit of logical thinking and some basic arithmetic, but don't worry, we'll make it super clear!

Defining the Terms

Before we jump into solving, let's define some key terms to make sure we're all on the same page:

  • Natural Numbers (N): These are the positive whole numbers starting from 1 (1, 2, 3, ...). Sometimes, depending on the context, 0 might be included, but for this problem, we'll stick to positive integers.
  • Divisible by 5: A number is divisible by 5 if it can be divided by 5 with no remainder. In other words, the result of the division is a whole number. For example, 10, 15, 20, and 25 are all divisible by 5.
  • x < 10: This inequality means that 'x' can be any natural number less than 10. So, x can be 1, 2, 3, 4, 5, 6, 7, 8, or 9.

Setting Up the Condition

Our main condition is that 5 must divide (15 + x). This can be written mathematically as:

5 | (15 + x)

This notation means that (15 + x) is a multiple of 5. So, we need to find the values of x (less than 10) that make (15 + x) a multiple of 5. Understanding this condition is crucial because it sets the stage for our problem-solving approach. We're essentially looking for 'x' values that, when added to 15, give us numbers like 15, 20, 25, and so on.

Finding the Values of x

Now, let's roll up our sleeves and find those values of 'x'! We know 'x' must be a natural number less than 10, and (15 + x) must be divisible by 5. We'll use a systematic approach, testing each possible value of x and checking if it meets our divisibility condition. This method might seem straightforward, but it's incredibly effective for problems like this.

Testing Possible Values

We'll go through each natural number less than 10 and see if adding it to 15 gives us a multiple of 5. Here’s how we do it:

  1. If x = 1:

    • 15 + x = 15 + 1 = 16
    • 16 is not divisible by 5 (16 ÷ 5 = 3 with a remainder of 1)

  2. If x = 2:

    • 15 + x = 15 + 2 = 17
    • 17 is not divisible by 5 (17 ÷ 5 = 3 with a remainder of 2)

  3. If x = 3:

    • 15 + x = 15 + 3 = 18
    • 18 is not divisible by 5 (18 ÷ 5 = 3 with a remainder of 3)

  4. If x = 4:

    • 15 + x = 15 + 4 = 19
    • 19 is not divisible by 5 (19 ÷ 5 = 3 with a remainder of 4)

  5. If x = 5:

    • 15 + x = 15 + 5 = 20
    • 20 is divisible by 5 (20 ÷ 5 = 4 with no remainder)

  6. If x = 6:

    • 15 + x = 15 + 6 = 21
    • 21 is not divisible by 5 (21 ÷ 5 = 4 with a remainder of 1)

  7. If x = 7:

    • 15 + x = 15 + 7 = 22
    • 22 is not divisible by 5 (22 ÷ 5 = 4 with a remainder of 2)

  8. If x = 8:

    • 15 + x = 15 + 8 = 23
    • 23 is not divisible by 5 (23 ÷ 5 = 4 with a remainder of 3)

  9. If x = 9:

    • 15 + x = 15 + 9 = 24
    • 24 is not divisible by 5 (24 ÷ 5 = 4 with a remainder of 4)

  10. If x = 0:

    • 15 + x = 15 + 0 = 15
    • 15 is divisible by 5 (15 ÷ 5 = 3 with no remainder)

Identifying the Solutions

From our tests, we found that:

  • When x = 5, 15 + x = 20, which is divisible by 5.
  • When x = 0, 15 + x = 15, which is divisible by 5.

So, the values of x that satisfy the condition are 5 and 0.

Expressing the Solution

Alright, we've done the hard work and found the values of 'x'! Now, let's clearly state our solution. The problem asked us to find the values of x in the set of natural numbers (N), where x is less than 10, such that 5 divides (15 + x). We tested all the natural numbers less than 10 and identified those that meet this condition.

The Solution Set

Based on our calculations, the values of x that satisfy the given conditions are:

  • x = 5
  • x = 0

So, we can express our solution as a set:

{0, 5}

This set includes all the natural numbers less than 10 that, when added to 15, result in a number divisible by 5. It's important to present the solution in a clear and concise manner, so anyone looking at our answer can easily understand it.

Verification

To be absolutely sure we've got it right, let's quickly verify our solutions. We'll plug each value of x back into the original condition and check if it holds true.

  1. For x = 5:

    • 15 + x = 15 + 5 = 20
    • 20 is divisible by 5 (20 ÷ 5 = 4)
    • So, x = 5 works!

  2. For x = 0:

    • 15 + x = 15 + 0 = 15
    • 15 is divisible by 5 (15 ÷ 5 = 3)
    • So, x = 0 works!

Both values, 5 and 0, satisfy the condition, confirming that our solution set is correct. This step-by-step verification process is a great habit to get into, especially in math, as it helps ensure accuracy and builds confidence in your answer.

Importance of Divisibility

Divisibility is a fundamental concept in mathematics, and it's used in various areas, from basic arithmetic to more advanced topics like number theory and cryptography. Understanding divisibility helps us simplify fractions, find common factors, and solve many types of mathematical problems. This problem, while simple, highlights the practical application of divisibility rules.

Real-World Applications

Divisibility isn't just a theoretical concept; it has real-world applications too. For example:

  • Scheduling: If you're planning an event and need to divide attendees into equal groups, divisibility helps ensure that everyone is accommodated properly.
  • Resource Allocation: In logistics, divisibility is crucial for distributing resources evenly. For instance, if you have a certain number of items to pack into boxes, you need to know if the number of items is divisible by the number of boxes to avoid leftovers.
  • Computer Science: Divisibility plays a key role in algorithms and data structures, especially in areas like hashing and modular arithmetic.

Tips for Mastering Divisibility

If you want to improve your understanding of divisibility, here are a few tips:

  • Learn the divisibility rules: Knowing the rules for divisibility by 2, 3, 4, 5, 6, 9, and 10 can save you a lot of time. For example, a number is divisible by 5 if its last digit is 0 or 5.
  • Practice, practice, practice: The more you work with divisibility problems, the better you'll become at recognizing patterns and applying the rules.
  • Use real-world examples: Try to relate divisibility to everyday situations. This can make the concept more tangible and easier to grasp.

Conclusion

So, there you have it, guys! We successfully found the values of x that satisfy the condition 5 divides (15 + x), where x is a natural number less than 10. The solutions are x = 5 and x = 0. By systematically testing each possible value and verifying our answers, we've reinforced our understanding of divisibility and problem-solving techniques.

Remember, math isn't just about getting the right answer; it's about the process of thinking logically and methodically. Keep practicing, keep exploring, and you'll be amazed at how much you can achieve. Until next time, happy problem-solving!