Linear Combination of GCD Calculator

Example Data Table

Formula Used

The calculator uses Bézout’s identity:

gcd(a, b) = ax + by

Here, a and b are integers. The values x and y are integer coefficients found by the extended Euclidean algorithm.

For a target equation:

aX + bY = c

A solution exists only when:

gcd(a, b) divides c

If one solution is X0 and Y0, then all integer solutions are:

X = X0 + (b / gcd(a,b))n

Y = Y0 - (a / gcd(a,b))n

How to Use This Calculator

1. Enter the first integer in the field marked a.
2. Enter the second integer in the field marked b.
3. Enter a target value c if you want to solve ax + by = c.
4. Leave the target blank if you only need the gcd combination.
5. Enter a parameter n to test one general solution pair.
6. Press the calculate button.
7. Read the result shown above the form.
8. Download the CSV or PDF report when needed.

Understanding Linear Combination of GCD

A linear combination of gcd shows how two integers build their greatest common divisor. Bézout’s identity says that, for integers a and b, there are integers x and y where ax + by equals gcd(a,b). This fact is central in number theory. It also supports modular inverses, Diophantine equations, cryptography exercises, and divisibility proofs.

Why coefficients matter

The coefficients x and y are not random. They come from reversing the Euclidean algorithm. Each division step writes a remainder using earlier remainders. When the process reaches the gcd, the tool substitutes backward and returns one valid pair. Different valid pairs exist, because adding b divided by the gcd to x and subtracting a divided by the gcd from y keeps the same value.

Practical uses

Linear combinations help solve equations such as ax + by = c. A solution exists only when the gcd of a and b divides c. If that condition is true, one Bézout pair can be scaled by c divided by the gcd. The calculator also shows the general integer solution. This helps learners see every possible pair instead of only one answer.

Learning with steps

The Euclidean table is useful for checking work. It shows each quotient, dividend, divisor, and remainder. These rows explain why the gcd is correct. The coefficient table then connects the final divisor with the original inputs. This makes the answer transparent and easier to verify.

Good input habits

Use integers, including negative values when needed. Very large values may be limited by the server integer range. For classroom work, choose values with several division steps. They make the method easier to observe. For equation solving, enter a target value. Then compare the sample solution with the general form. This gives a complete view of the relationship between divisibility, gcd, and integer equations.

Accuracy notes

The result line should always multiply back to the displayed gcd. If the target equation is solved, the scaled pair should multiply back to the target. When no solution appears, the divisibility test failed. That failure is not an error. It means the requested value cannot be created from the two inputs using integer coefficients. It also prevents misleading decimal answers in exact arithmetic work.

FAQs