GCD Of 14n+9 And 25n+4: Finding The Maximum Value

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Hey guys! Let's dive into an interesting math problem today. We're going to explore how to find the largest possible value for the greatest common divisor (GCD) of two expressions: 14n + 9 and 25n + 4, where n is a natural number. This is a classic number theory problem that involves some clever algebraic manipulation and a good understanding of GCD properties. So, buckle up and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the problem is asking. The greatest common divisor (GCD), also known as the highest common factor (HCF), of two numbers is the largest positive integer that divides both numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18.

In this problem, we're not dealing with fixed numbers, but rather with two expressions, 14n + 9 and 25n + 4, that depend on the natural number n. A natural number is a positive whole number (1, 2, 3, and so on). Our goal is to find the largest possible number that can divide both 14n + 9 and 25n + 4 for some value of n.

This means we need to find a common divisor that works for any n. To tackle this, we'll use some properties of the GCD and some algebraic tricks to simplify the expressions.

The Euclidean Algorithm and GCD Properties

The Euclidean Algorithm is a fundamental method for finding the GCD of two numbers. It's based on the principle that the GCD of two numbers doesn't change if the larger number is replaced by its difference with the smaller number. We can extend this idea to our expressions. A crucial property we'll use is:

  • GCD(a, b) = GCD(a, b - ka) for any integer k.

This property states that we can subtract a multiple of one number from the other without changing their GCD. This is incredibly useful because it allows us to eliminate the n term and find a constant GCD.

Solving the Problem: A Step-by-Step Approach

Let's denote the GCD of 14n + 9 and 25n + 4 as d. So,

  • d = GCD(14n + 9, 25n + 4)

Our goal is to manipulate these expressions to eliminate n and find a constant value for d. Here’s how we can do it:

  1. Multiply the expressions to align the 'n' coefficients:

    • Multiply the first expression (14n + 9) by 25: 25(14n + 9) = 350n + 225
    • Multiply the second expression (25n + 4) by 14: 14(25n + 4) = 350n + 56

    Now we have two new expressions with the same coefficient for n (350n). This is a crucial step because it allows us to eliminate n by subtracting the expressions.

  2. Subtract the expressions:

    • Subtract the second expression from the first: (350n + 225) - (350n + 56) = 169

    Notice that the n term has disappeared! We're left with a constant, 169. This is a huge breakthrough.

  3. Apply the GCD property:

    • Since d divides both 14n + 9 and 25n + 4, it must also divide any linear combination of them. In particular, it must divide the difference we just calculated:
    • d | 169 (where '|' means 'divides')

    This tells us that d is a divisor of 169. Now we need to find the largest possible divisor of 169.

  4. Find the divisors of 169:

    • The divisors of 169 are 1, 13, and 169 (since 169 = 13 * 13).

    So, the possible values for d are 1, 13, and 169. The largest possible value is 169. But can we actually achieve this value?

  5. Check if the maximum GCD is achievable:

    • We need to find an n such that 169 divides both 14n + 9 and 25n + 4. Let's set up the following conditions:

      • 14n + 9 ≡ 0 (mod 169)
      • 25n + 4 ≡ 0 (mod 169)

    • From the first congruence, we have 14n ≡ -9 (mod 169), which is equivalent to 14n ≡ 160 (mod 169).

    • From the second congruence, we have 25n ≡ -4 (mod 169), which is equivalent to 25n ≡ 165 (mod 169).

    • To solve these congruences, we can use the modular inverse. However, let's try a simpler approach first. Divide the second congruence by 5: 5n ≡ 33 (mod 169).

    • Now, multiply this new congruence by 14: 70n ≡ 462 (mod 169).

    • Reduce 462 modulo 169: 462 = 2 * 169 + 124, so 70n ≡ 124 (mod 169).

    • Multiply the first original congruence (14n ≡ 160 (mod 169)) by 5: 70n ≡ 800 (mod 169).

    • Reduce 800 modulo 169: 800 = 4 * 169 + 144, so 70n ≡ 144 (mod 169).

    • Now we have two congruences:

      • 70n ≡ 124 (mod 169)
      • 70n ≡ 144 (mod 169)

    • Since the left sides are the same, the right sides must be congruent: 124 ≡ 144 (mod 169). This is not true, but it doesn't mean a solution doesn't exist. It means we might need to adjust our approach slightly.

    • Let's go back to the original congruences and try a different tack. We have:

      • 14n + 9 = 169k for some integer k
      • 25n + 4 = 169m for some integer m

    • Multiply the first equation by 25 and the second by 14:

      • 350n + 225 = 4225k
      • 350n + 56 = 2366m

    • Subtract the second equation from the first:

      • 169 = 4225k - 2366m

    • Divide by 169: 1 = 25k - 14m

    • This is a linear Diophantine equation. We can find a solution using the extended Euclidean algorithm or by inspection. One solution is k = 1 and m = (25 - 1)/14 = 24/14, which is not an integer. Let's try another approach. We need to find integers k and m such that 25k - 14m = 1. Using the extended Euclidean algorithm or inspection, one solution is k=1, m=1. Thus, 25(1) - 14(1) = 11. Another solution is k = 3, m = 5, so 25(3) - 14(5) = 75 - 70 = 5. After trying a bit more, we find k = 11, m = 19. 25(11) - 14(19) = 275 - 266 = 9. Still no solution. We should use Extended Euclidean Algorithm. Extended Euclidean Algorithm for 25 and 14: 25 = 1 * 14 + 11 14 = 1 * 11 + 3 11 = 3 * 3 + 2 3 = 1 * 2 + 1

      1 = 3 - 1 * 2 1 = 3 - 1 * (11 - 3 * 3) = 4 * 3 - 1 * 11 1 = 4 * (14 - 1 * 11) - 1 * 11 = 4 * 14 - 5 * 11 1 = 4 * 14 - 5 * (25 - 1 * 14) = 9 * 14 - 5 * 25 So 1 = 9 * 14 - 5 * 25. Therefore -5*25 + 9 * 14 = 1

    • Let k = -5 and m= -9. Substituting k=9 into 14n + 9 = 169k. 14n + 9 = 169*9 = 1521. 14n = 1512. n=108 Then 25(108) + 4 = 2700 + 4 = 2704 GCD (1521, 2704)= 169. So n = 108 is a solution.

The Final Answer

Therefore, the largest possible value for the greatest common divisor of 14n + 9 and 25n + 4 is 169. This occurs when n = 108 (and potentially other values as well).

Key Takeaways

  • The Euclidean Algorithm and GCD properties are powerful tools for solving number theory problems.
  • Manipulating expressions to eliminate variables is a common strategy for finding GCDs.
  • Checking if a potential maximum GCD is achievable is crucial to ensure the solution is valid.

I hope this explanation was helpful, guys! Remember, practice makes perfect. The more you work on these types of problems, the better you'll become at solving them. Keep exploring the fascinating world of number theory!