Finding Coprime Numbers: Solving 3a + 4b = 39 And 3a + 2b = 54

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Finding Coprime Numbers: Solving 3a + 4b = 39 and 3a + 2b = 54

Hey guys! Let's dive into some cool math problems today! We're going to tackle two equations and find coprime numbers that fit them. Coprime numbers, as you might remember, are numbers that have only 1 as their common factor. So, no shared prime factors, got it? Let's get started!

Determining Coprime Numbers for 3a + 4b = 39

So, the first task is to find coprime numbers a and b that satisfy the equation 3a + 4b = 39. This looks like a fun puzzle, right? The key here is to think about how we can manipulate the equation and what kind of numbers will fit. First off, we know that a and b have to be whole numbers (positive integers) because we're talking about coprime numbers. We can start by rearranging the equation to isolate one of the variables. Let's isolate a:

3a = 39 - 4b a = (39 - 4b) / 3

Now, we need to think about what values of b will make a a whole number. This means that (39 - 4b) must be divisible by 3. Let's try some values for b and see what happens.

If b = 1, then a = (39 - 4) / 3 = 35 / 3, which isn't a whole number. Nope! If b = 2, then a = (39 - 8) / 3 = 31 / 3, still not a whole number. Keep trying! If b = 3, then a = (39 - 12) / 3 = 27 / 3 = 9. Bingo! We have a potential solution: a = 9 and b = 3. But wait, are they coprime? The factors of 9 are 1, 3, and 9, and the factors of 3 are 1 and 3. They share a common factor of 3, so they are not coprime. Bummer. If b = 4, then a = (39 - 16) / 3 = 23 / 3, not a whole number. If b = 5, then a = (39 - 20) / 3 = 19 / 3, still not a whole number. If b = 6, then a = (39 - 24) / 3 = 15 / 3 = 5. Okay, a = 5 and b = 6. Let's check if they are coprime. The factors of 5 are 1 and 5, and the factors of 6 are 1, 2, 3, and 6. They only share the factor 1, so a = 5 and b = 6 are coprime! We found a solution! If we continue trying larger values for b, a becomes negative, and we're looking for positive integers, so we can stop here.

So, the coprime numbers that satisfy 3a + 4b = 39 are a = 5 and b = 6. Isn't that cool how we figured it out step by step? We used a bit of trial and error, but by understanding the equation and what coprime means, we cracked the code!

Determining Coprime Numbers for 3a + 2b = 54

Alright, let's move on to the next challenge! We need to find coprime numbers a and b that fit the equation 3a + 2b = 54. Ready for round two? We'll use a similar approach as before. First, we isolate a:

3a = 54 - 2b a = (54 - 2b) / 3

Now we need to find values of b that make (54 - 2b) divisible by 3, ensuring that a is a whole number. Let's plug in some values for b and see what we get:

If b = 1, then a = (54 - 2) / 3 = 52 / 3, not a whole number. Next! If b = 2, then a = (54 - 4) / 3 = 50 / 3, still not a whole number. If b = 3, then a = (54 - 6) / 3 = 48 / 3 = 16. So, a = 16 and b = 3. Are they coprime? The factors of 16 are 1, 2, 4, 8, and 16, and the factors of 3 are 1 and 3. They only share the factor 1, so a = 16 and b = 3 are coprime! Awesome! If b = 6, then a = (54 - 12) / 3 = 42 / 3 = 14. So, a = 14 and b = 6. The factors of 14 are 1, 2, 7, and 14, and the factors of 6 are 1, 2, 3, and 6. They share the factors 1 and 2, so they are not coprime. Nice try, though! If b = 9, then a = (54 - 18) / 3 = 36 / 3 = 12. So, a = 12 and b = 9. The factors of 12 are 1, 2, 3, 4, 6, and 12, and the factors of 9 are 1, 3, and 9. They share the factors 1 and 3, so they are not coprime. Keep going! If b = 12, then a = (54 - 24) / 3 = 30 / 3 = 10. So, a = 10 and b = 12. The factors of 10 are 1, 2, 5, and 10, and the factors of 12 are 1, 2, 3, 4, 6, and 12. They share the factors 1 and 2, so they are not coprime. If b = 15, then a = (54 - 30) / 3 = 24 / 3 = 8. So, a = 8 and b = 15. The factors of 8 are 1, 2, 4, and 8, and the factors of 15 are 1, 3, 5, and 15. They only share the factor 1, so a = 8 and b = 15 are coprime! We found another solution! If b = 18, then a = (54 - 36) / 3 = 18 / 3 = 6. So, a = 6 and b = 18. They clearly share more than just the factor 1 (like 6), so they are not coprime. If b = 21, then a = (54 - 42) / 3 = 12 / 3 = 4. So, a = 4 and b = 21. The factors of 4 are 1, 2, and 4, and the factors of 21 are 1, 3, 7, and 21. They only share the factor 1, so a = 4 and b = 21 are coprime! If b = 24, then a = (54 - 48) / 3 = 6 / 3 = 2. So, a = 2 and b = 24. They share the factor 2, so they are not coprime. If b = 27, then a = (54 - 54) / 3 = 0 / 3 = 0. While 0 is an integer, coprime numbers are generally considered to be positive integers, so we stop here.

So, the coprime numbers that satisfy 3a + 2b = 54 are a = 16, b = 3; a = 8, b = 15; and a = 4, b = 21. We nailed it! We found multiple solutions for this one.

Key Takeaways

  • Understanding Coprime Numbers: Remember, coprime numbers only share the factor 1. This is crucial for solving these types of problems.
  • Rearranging Equations: Isolating one variable helps us see the relationship between the numbers more clearly.
  • Trial and Error: Don't be afraid to try different values! It's a great way to learn and find solutions.
  • Checking for Coprime: Always verify that your solutions are indeed coprime by listing their factors.

Conclusion

Math can be a fun adventure, and finding coprime numbers is like solving a cool puzzle. By using a combination of algebraic manipulation and a bit of trial and error, we can crack these problems. Keep practicing, and you'll become a coprime-finding pro in no time! Keep exploring the world of numbers, guys! There's always something new and exciting to discover.