Final answer:
To find the positive integer solutions to the equation 13x + 15y = 450, use the method of linear Diophantine equations and the extended Euclidean algorithm. The general equation for the solutions is x = 900 + 15n, y = -450 + 13n, where n is a positive integer.
Step-by-step explanation:
To find the positive integer solutions to the equation 13x + 15y = 450, we can use the method of linear Diophantine equations. First, we need to find the greatest common divisor (GCD) of 13 and 15, which is 1. Since 1 divides 450, there are infinite solutions to this equation.
To find one particular solution, we can use the extended Euclidean algorithm. By applying the extended Euclidean algorithm to 13 and 15, we get:
13(2) - 15(1) = 1.
Multiplying both sides of the equation by 450 gives:
13(2)(450) - 15(1)(450) = 450.
This yields one particular solution: x = 2(450) = 900; y = -1(450) = -450.
Since the GCD of 13 and 15 is 1, we can add multiples of 15 to x and multiples of 13 to y to obtain more solutions. The general equation for the positive integer solutions is:
x = 900 + 15n,
y = -450 + 13n,
where n is a positive integer.