Okay, let's find all integer solutions to the Diophantine equation 3x + 5y = 100. **1. Finding a Particular Solution** We can use the Extended Euclidean Algorithm or simply observation to find a particular solution. Since 100 is divisible by 5, it's easiest to find a solution where x is a multiple of 5. Let's try x = 0. Then 3(0) + 5y = 100, so 5y = 100, and y = 20. Thus, (x₀, y₀) = (0, 20) is a particular solution. **2. Finding the General Solution** Now, let (x, y) be any other integer solution to the equation 3x + 5y = 100. Since (x₀, y₀) = (0, 20) is also a solution, we have: 3x + 5y = 100 3(0) + 5(20) = 100 Subtracting the two equations: 3x + 5y - (3(0) + 5(20)) = 100 - 100 3x + 5y - 5(20) = 0 3x + 5(y - 20) = 0 3x = -5(y - 20) Since 3 and 5 are relatively prime (gcd(3, 5) = 1), it must be the case that x is a multiple of 5, and (y-20) is a multiple of 3. Let x = 5k for some integer k. Substituting this into the equation: 3(5k) = -5(y - 20) 15k = -5(y - 20) 3k = -(y - 20) 3k = -y + 20 y = 20 - 3k Therefore, the general solution is given by: x = 5k y = 20 - 3k where k is any integer. **3. Listing All Integer Solutions** We want to find all integer pairs (x, y) that satisfy 3x + 5y = 100. Since x and y must be non-negative, we must solve for valid k. x >= 0 => 5k >= 0 => k >= 0 y >= 0 => 20 - 3k >= 0 => 20 >= 3k => k <= 20/3 => k <= 6.66... So k must be an integer between 0 and 6 inclusive. Let's find the corresponding x and y values for each value of k: * k = 0: x = 5(0) = 0, y = 20 - 3(0) = 20. (0, 20) * k = 1: x = 5(1) = 5, y = 20 - 3(1) = 17. (5, 17) * k = 2: x = 5(2) = 10, y = 20 - 3(2) = 14. (10, 14) * k = 3: x = 5(3) = 15, y = 20 - 3(3) = 11. (15, 11) * k = 4: x = 5(4) = 20, y = 20 - 3(4) = 8. (20, 8) * k = 5: x = 5(5) = 25, y = 20 - 3(5) = 5. (25, 5) * k = 6: x = 5(6) = 30, y = 20 - 3(6) = 2. (30, 2) Therefore, the integer solutions are: (0, 20), (5, 17), (10, 14), (15, 11), (20, 8), (25, 5), (30, 2).

Integer solutions to 3x + 5y = 100

gemini/gemini-2.0-flash

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