Bezout Identity Calculator
Find integers x and y satisfying Bezout's identity: ax + by = gcd(a, b).
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Bezout's Identity Theorem
For any integers a and b (not both zero), there exist integers x and y such that:
ax + by = gcd(a, b)
The coefficients x and y are called Bezout coefficients.
Properties
- Bezout coefficients are not unique
- If (x₀, y₀) is a solution, all solutions are:
- x = x₀ + (b/d)k
- y = y₀ - (a/d)k
- where d = gcd(a, b) and k is any integer
Bezout Identity
35(1) + 15(-2) = 5
GCD
5
x
1
y
-2
General Solution
x = 1 + 3k
y = -2 - 7k
where k is any integer
Sample Coefficient Pairs
| k | x | y | ax+by |
|---|---|---|---|
| -5 | -14 | 33 | 5 |
| -4 | -11 | 26 | 5 |
| -3 | -8 | 19 | 5 |
| -2 | -5 | 12 | 5 |
| -1 | -2 | 5 | 5 |
| 0 | 1 | -2 | 5 |
| 1 | 4 | -9 | 5 |
| 2 | 7 | -16 | 5 |
| 3 | 10 | -23 | 5 |
| 4 | 13 | -30 | 5 |
| 5 | 16 | -37 | 5 |
Verification
35 × (1) + 15 × (-2) = 5
Identity verified