Finding The Sum: Relatively Prime Numbers With A Twist!

by Dimemap Team 56 views

Hey math whizzes and number enthusiasts! Let's dive into a fun problem that combines two-digit numbers, the concept of relatively prime numbers, and a bit of a puzzle. This one's all about strategy and a solid understanding of prime numbers, so buckle up! The core challenge is this: We've got a set of cards marked with '12 ∆3', and we're tasked with writing two-digit natural numbers on them. The kicker? The number on the first floor and the number on the second floor must be relatively prime (or coprime) to each other. Our ultimate goal is to figure out the sum of all possible digits that can replace the triangle symbol (∆). Let's break down this problem, step by step, and figure out how to crack it.

Unveiling the Problem's Core: Relatively Prime Numbers Explained

Alright, before we get our hands dirty with the actual problem, let's make sure we're on the same page when it comes to relatively prime numbers. Two numbers are considered relatively prime (or coprime) if their greatest common divisor (GCD) is 1. In simpler terms, this means that the only positive integer that divides both numbers without leaving a remainder is 1. For example, the numbers 7 and 10 are relatively prime because their only common divisor is 1. On the other hand, 6 and 9 are not relatively prime because they share a common divisor of 3 (besides 1). This basic understanding is absolutely crucial for solving this problem, so ensure you understand the key concept. The whole problem revolves around this single idea.

Now, let's translate this into our problem. We'll be working with two-digit numbers, which means our numbers will range from 10 to 99. The first number will be made up of the digits 1 and 2, and the second number will be composed of the digits 3 and whatever digit replaces the triangle symbol. Therefore, any solution must focus on relatively prime pairs within this specific constraint. This means that after forming a number on both cards, you must check their GCD, which must be 1. It is important to know that for two numbers to be relatively prime, they do not need to be prime numbers themselves. They only need to have 1 as their greatest common divisor.

The Strategy: Finding Compatible Digits

Now, how do we solve this? The trick is to systematically figure out which digits can replace the triangle symbol (∆) to ensure the two resulting numbers are relatively prime. Here's a structured approach:

  1. Understand the Fixed Number: We know the first number will always start with 12. We need to focus on this number as a constant to determine the second. Therefore, whatever digit we add to the second card must meet the requirement of being relatively prime to 12. This becomes our base, and everything will revolve around it. The main idea here is to not get confused by trying to work with all the possibilities at once. It's much easier to work around the number 12 and then verify the second number against it.
  2. Prime Factorization of the Fixed Number: Prime factorization is your best friend here. The prime factorization of 12 is 2 x 2 x 3, or more simply, 2² x 3. This tells us that 12 is divisible by 2 and 3. This crucial piece of information helps us determine which digits cannot be used in the second number. In other words, to ensure that the two numbers are relatively prime, the second number cannot be divisible by either 2 or 3. If it were, the GCD wouldn't be 1.
  3. Test Possible Second Numbers: Since the second number will start with 3, we'll test the possibilities for the second digit (replacing the triangle). The number options are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. We will create all possible combinations and evaluate which ones are relatively prime with the first number (12).
  4. Eliminate the Conflicts: Remember, we can't have any common factors with 12 except 1. This means the second number cannot be even (because 12 is even) or divisible by 3 (because 12 is divisible by 3). Let's break this down:
    • Even Numbers: Any number ending in 0, 2, 4, 6, or 8 is even and therefore shares a factor of 2 with 12. So, we immediately eliminate those possibilities.
    • Divisibility by 3: We also need to eliminate any numbers divisible by 3. This leaves us with a few more to exclude. If you are unsure whether a number is divisible by 3, you can add all the digits, and if the result is divisible by 3, then so is the number itself. For example, 33 is divisible by 3 because 3 + 3 = 6, and 6 is divisible by 3.

Putting it All Together: The Solution

Let's apply these steps.

  1. Possible Digits: We start by considering the digits 0 through 9 to replace the triangle.
  2. Form the Numbers: The first number is always 12. The second number will be 30, 31, 32, 33, 34, 35, 36, 37, 38, or 39, depending on which digit replaces the triangle.
  3. Test for Coprimality: Let's look at each possible second number and see if it is relatively prime with 12:
    • 30: Not relatively prime (both divisible by 2 and 3).
    • 31: Relatively prime (only divisible by 1 and 31).
    • 32: Not relatively prime (both divisible by 2).
    • 33: Not relatively prime (both divisible by 3).
    • 34: Not relatively prime (both divisible by 2).
    • 35: Relatively prime (only divisible by 5 and 7).
    • 36: Not relatively prime (both divisible by 2 and 3).
    • 37: Relatively prime (only divisible by 1 and 37).
    • 38: Not relatively prime (both divisible by 2).
    • 39: Not relatively prime (both divisible by 3).
  4. Identify Valid Digits: The digits that work are those that result in a number relatively prime to 12. In our case, the digits are 1, 5, and 7.
  5. Calculate the Sum: Finally, we add these digits together: 1 + 5 + 7 = 13. Therefore, the sum of the possible digits that can replace the triangle symbol is 13. This process, while seemingly long, guarantees we consider all valid possibilities and arrive at the correct answer.

Therefore, the answer is not among the options given. The correct answer is 13, and the options are all incorrect. If there was a different number in the first card, this could change. However, based on the question, the result is 13.

Diving Deeper: Further Exploration

Now, let's explore this concept a bit further. What if the first number was different? For example, let's say the first number was 15. The prime factorization of 15 is 3 x 5. Therefore, the second number cannot be divisible by 3 or 5. This would change our approach completely. The numbers would be 30, 31, 32, 33, 34, 35, 36, 37, 38, or 39. Then we would eliminate all numbers divisible by 3 or 5. 30 (divisible by 3 and 5), 31 (relatively prime), 32 (relatively prime), 33 (divisible by 3), 34 (relatively prime), 35 (divisible by 5), 36 (divisible by 3), 37 (relatively prime), 38 (relatively prime), and 39 (divisible by 3). The valid digits would be 1, 2, 4, 7, and 8. If we add them, the sum is 22.

The same process can be applied to different problems, changing the prime factorization, numbers, and answers. Knowing prime factorization is the key to solving this. You can also reverse engineer this approach. If you are given the solution, you can find the prime factorization and work from there.

Conclusion: Mastering the Concept

We've successfully navigated this problem, learned about relatively prime numbers, and used prime factorization to find the answer. Remember, the core concept lies in understanding the definition of relatively prime numbers and how prime factorization helps identify common factors. By following a structured approach, like the one we used, you can confidently tackle these types of problems. Keep practicing, and you'll become a number-crunching pro in no time! So, keep exploring, keep questioning, and most importantly, keep enjoying the world of numbers!