basic

Linear Diophantine Equation Integer Solution Calculator

ax + by = c — general integer solution from particular solution.

Given a particular integer solution (x₀, y₀) and GCD d of a and b, computes the general solution: x = x₀ + (b/d)t, y = y₀ − (a/d)t for any integer t.

Marcus Laurent By Marcus Laurent
Published Last reviewed 1 min read

Inputs

Results

Enter values and click Calculate to see results.
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How to use this calculator

  1. Fill in the inputs above using the units you already have.
  2. Values update automatically as you type — no submit button needed.
  3. Hover any result row for the underlying formula and intermediate values.

Formula

x(t) = x₀ + (b/d)·t; y(t) = y₀ − (a/d)·t; valid if d | c

In depth

Given a particular integer solution (x₀, y₀) and GCD d of a and b, computes the general solution: x = x₀ + (b/d)t, y = y₀ − (a/d)t for any integer t.