Understanding the Number of Positive Integral Solutions to Linear Diophantine Equations

Linear Diophantine equations are a fascinating area of number theory, providing a framework for finding integer solutions to equations of the form ax by c. In this article, we will delve into specific cases, beginning with a detailed exploration of the equation 2x 3y 600. We will not only derive the number of positive integral solutions but also provide a general theorem to handle such problems efficiently.

Specific Case: 2x 3y 600

To find the number of positive integral solutions for the equation 2x 3y 600, we can follow a structured approach:

Rearranging the equation: We start by expressing y in terms of x.

3y 600 - 2x implies y frac{600 - 2x}{3}

Conditions for y to be an integer: For y to be an integer, 600 - 2x must be divisible by 3. Since 600 3 times 200, the condition simplifies to 2x equiv 0 pmod{3}, which implies x equiv 0 pmod{3}. Thus, x must be a multiple of 3.

Substitution: Let x 3k for some positive integer k. Substituting this into the equation gives:

2(3k) 3y 600 implies 3y 600 - 6k implies y 200 - 2k

Finding bounds for k: Since both x and y must be positive integers, these conditions must hold:

x 3k 0 implies k 0 y 200 - 2k 0 implies k 100

Therefore, k can take any integer value from 1 to 99, inclusive.

Counting the solutions: The possible values of k are the integers from 1 to 99, giving us 99 solutions.

General Theorem on Linear Diophantine Equations

The problem of finding the number of positive integral solutions to equations of the form ax by c can be approached using a more general theorem:

Theorem. Let a, b, n in mathbb{N} and gcd(a, b) 1. The number of ordered pairs (x, y) in mathbb{N} satisfying ax by abn is n - 1.

Proof: If ax by abn, then a mid by and b mid ax. Since gcd(a, b) 1, it follows that a mid y and b mid x. Write x bx_1 and y ay_1 where x_1, y_1 in mathbb{N} and substitute in ax by abn. This gives x_1y_1 n. For x_1 and y_1) to be positive integers, it is necessary and sufficient that x_1 in {1, 2, ldots, n - 1}. Thus there are n - 1 solutions.

Linear Diophantine Equation: 2x 3y 600

Applying the theorem to the equation 2x 3y 600 with a 2, b 3, and n 600, we find that the number of positive integral solutions is 99.

General Theorem: Linear Diophantine Equations

To further generalize, the linear Diophantine equation ax by n has a solution in integers x and y if and only if gcd(a, b) mid n.

Proof: If ax by n has a solution, then ax_0 by_0 n for some integers x_0, y_0. This implies gcd(a, b) mid n. Conversely, if gcd(a, b) mid n, let d gcd(a, b), and write n dk. Then the equation a'x b'y k has a solution (where a' frac{a}{d} and b' frac{b}{d} since gcd(a', b') 1) and thus has infinitely many solutions in integers x_0, y_0. The general solution is given by x x_0 frac{b'}{d}t, y y_0 - frac{a'}{d}t for t in mathbb{Z}.

Conclusion

Understanding and solving linear Diophantine equations involves a combination of algebraic manipulation and number theory principles. By leveraging theorems and properties of greatest common divisors, we can efficiently determine the number of positive integral solutions. This approach not only provides a framework for specific instances but also offers a general method for tackling similar problems.