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Diophantine Equation Calculator

Solve linear Diophantine equations of the form ax + by = c. Find integer solutions using the Extended Euclidean Algorithm.

Inputs

Results

Solution Exists

Yes

x₀ (particular)2
y₀ (particular)-1
GCD(a, b)1
General Solutionx = 2 + 5t, y = -1 - 3t
How to Use This Calculator
  1. Enter a, b, and c.
  2. Review the Solution Exists result.
  3. Use x₀ (particular) and y₀ (particular) to inform your decision.
  4. Use the chart to visualize the results and explore different scenarios by adjusting inputs.

What each input means

a
Coefficient a in ax + by = c
b
Coefficient b in ax + by = c
c
Right-hand side constant c in ax + by = c

How this is calculated

Worked example, using the default values

  1. Identify Input Parameters
    a = 3, b = 5, c = 1 = 3 input(s) provided
  2. Calculate Solution Exists
    Yes = Yes
  3. Calculate x₀
    2 = 2
  4. Calculate y₀
    -1 = -1

Engine last updated . Checked against 3 independently-derived tests — how we verify calculators.

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