Find A And B: GCD(a, B) = 5 And A*b = 175

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Find a and b knowing that (a,b)=5 and a•b=175

Let's dive into solving this math problem together, guys! We're given two key pieces of information: the greatest common divisor (GCD) of two numbers, a and b, is 5, and their product, a times b, is 175. Our mission, should we choose to accept it, is to find the values of a and b. Buckle up; it's going to be a fun ride through number theory!

Understanding the Basics

Before we start crunching numbers, let's make sure we're all on the same page with the fundamental concepts. The greatest common divisor (GCD), also known as the highest common factor (HCF), is the largest positive integer that divides both numbers without leaving a remainder. In our case, GCD(a, b) = 5 tells us that 5 is the biggest number that divides both a and b evenly. This is super important because it gives us a starting point for expressing a and b. Furthermore, we know that a•b=175, meaning that when we multiply a and b we get 175. This condition will make it possible to correctly identify the a and b numbers.

Expressing a and b

Since 5 is the GCD of a and b, we can express a and b as multiples of 5. Let's say a = 5x and b = 5y, where x and y are integers. The crucial part here is that x and y must be coprime, meaning their GCD is 1. If x and y had a common factor greater than 1, then the GCD of a and b would be larger than 5, which contradicts our given information. Keep this in mind, it's a golden rule! This is a critical step in solving problems involving GCD, so remember it, because you'll use this technique often.

Using the Product Information

Now, let's use the fact that a * b* = 175. Substitute a = 5x and b = 5y into this equation:

(5x) * (5y) = 175

This simplifies to:

25xy = 175

Divide both sides by 25:

xy = 7

Finding x and y

Now we need to find two coprime integers x and y such that their product is 7. Since 7 is a prime number, its only factors are 1 and 7. Therefore, the possible pairs for (x, y) are (1, 7) and (7, 1). Remember, x and y must be coprime, and in this case, 1 and 7 are indeed coprime (their GCD is 1). Great! We're on the right track!

Calculating a and b

Now that we have the possible pairs for (x, y), we can find the corresponding values for a and b.

Case 1: (x, y) = (1, 7)

a = 5x = 5 * 1 = 5

b = 5y = 5 * 7 = 35

Case 2: (x, y) = (7, 1)

a = 5x = 5 * 7 = 35

b = 5y = 5 * 1 = 5

So, in both cases, we find that the numbers are 5 and 35. The order doesn't matter since the problem doesn't specify which number is a and which is b. Awesome!

Final Answer

Therefore, the numbers a and b are 5 and 35. We can verify this by checking that GCD(5, 35) = 5 and 5 * 35 = 175. Success! We solved the problem.

Additional Insights and Tips

Importance of Coprime Numbers

Always remember the condition that x and y must be coprime when expressing a and b in terms of their GCD. This is a critical step in ensuring that you find the correct solutions. Forgetting this can lead to incorrect answers. Always double-check!

Prime Factorization

Prime factorization can be a helpful tool in solving problems involving GCD and LCM (least common multiple). By breaking down numbers into their prime factors, you can easily identify common factors and determine the GCD. Although not strictly necessary for this particular problem, it's a good technique to have in your toolbox.

Alternative Approach: Listing Factors

Another approach, especially useful for smaller numbers, is to list the factors of 175 and identify pairs that have a GCD of 5.

Factors of 175: 1, 5, 7, 25, 35, 175

Possible pairs: (1, 175), (5, 35), (7, 25)

Now check the GCD of each pair:

  • GCD(1, 175) = 1
  • GCD(5, 35) = 5
  • GCD(7, 25) = 1

Only the pair (5, 35) has a GCD of 5, confirming our solution. This method is more intuitive but can be less efficient for larger numbers.

General Strategy for GCD and Product Problems

Here’s a general strategy you can apply to similar problems:

  1. Express the numbers a and b as multiples of their GCD: a = GCD * x, b = GCD * y, where x and y are coprime.
  2. Use the given product a * b* to find the product x * y*.
  3. Find pairs of coprime integers (x, y) that satisfy the product.
  4. Calculate a and b using these pairs.
  5. Verify your solution by checking the GCD and product conditions.

Practice Problems

To solidify your understanding, try these practice problems:

  1. Find a and b if GCD(a, b) = 3 and a * b* = 108.
  2. Find x and y if GCD(x, y) = 7 and x * y* = 245.
  3. Determine m and n if GCD(m, n) = 4 and m * n* = 192.

Work through these problems using the strategies we discussed. Don't hesitate to review the steps if you get stuck. Practice makes perfect! And remember, math can be fun if you approach it with the right attitude and strategies.

Conclusion

So, there you have it! We successfully determined the numbers a and b given their GCD and product. Remember the key steps: expressing the numbers as multiples of their GCD, using the product information to find coprime factors, and verifying your solution. With these tools in your arsenal, you'll be well-equipped to tackle similar problems in the future. Keep practicing, and you'll become a math whiz in no time! Happy problem-solving, guys!