Euclidean Algorithm GCD Calculator

Calculator Input

First Integer

Second Integer

Pair Label

Prime Factor Limit

Advanced Options

Show Bézout coefficients

Show prime factors

Check modular inverse

Report Notes

Formula Used

The Euclidean algorithm uses repeated division:

a = bq + r

Here, a is the dividend, b is the divisor, q is the quotient, and r is the remainder.

The key rule is:

gcd(a, b) = gcd(b, r)

The process continues until the remainder becomes zero. The last nonzero divisor is the greatest common divisor.

The calculator also uses:

LCM(a, b) = |a × b| ÷ GCD(a, b)

ax + by = gcd(a, b)

How to Use This Calculator

Enter the first integer in the first input box.
Enter the second integer in the second input box.
Add a label if you want a named report.
Choose whether to show coefficients, factors, and inverse checks.
Press the calculate button.
Review the result shown above the form.
Use the CSV or PDF button to export the result.

Example Data Table

First Integer	Second Integer	GCD	LCM	Coprime
252	198	18	2772	No
1071	462	21	23562	No
35	64	1	2240	Yes
144	89	1	12816	Yes

Why This GCD Tool Helps

The Euclidean algorithm is one of the oldest useful methods in mathematics. It finds the greatest common divisor of two integers by repeated division. Each division replaces the larger value with the previous remainder. The process ends when the remainder becomes zero. The last nonzero remainder is the GCD. This calculator shows that full trail, so the result is easier to check.

Clear Number Theory Output

A basic GCD answer is often not enough. Students, teachers, and developers may need the quotient, remainder, LCM, prime factors, and Bézout coefficients. This page provides those values in one place. It also checks whether the two integers are coprime. When the GCD is one, the tool can show a modular inverse where a valid modulus exists.

Support For Detailed Work

The calculator accepts positive, negative, and zero values. It uses absolute values for the division chain, while the identity uses signs from the original input. This keeps the method mathematically consistent. The step table is helpful for homework, coding checks, cryptography practice, fraction reduction, and divisibility studies. The export buttons make it simple to keep a record.

Practical Study Benefits

Euclid’s method is fast because every remainder is smaller than the previous divisor. Large values usually shrink quickly. This is why the algorithm is still used in modern software. It is also used inside fraction tools, rational simplifiers, public key examples, and modular arithmetic exercises. By viewing each row, you can understand why the final divisor divides both starting numbers.

Best Use Cases

Use this calculator when you need proof, not only a final number. It is useful for reducing ratios, simplifying fractions, comparing periods, checking shared factors, and building number theory lessons. The example table gives quick practice values. The formula section explains the rule behind each division. The guide section explains the steps in plain language.

Accuracy Notes

For classroom use, enter integers exactly as given. For programming use, test small and large pairs. If one value is zero, the GCD is the absolute value of the other value. If both values are zero, no greatest common divisor exists. Always review the remainders before using exported reports. This keeps answers clear, traceable, and easy to audit.

FAQs

1. What is the Euclidean algorithm?

It is a repeated division method for finding the greatest common divisor of two integers. The method keeps replacing the larger value with a remainder until the remainder becomes zero.

2. What does GCD mean?

GCD means greatest common divisor. It is the largest positive integer that divides both given integers without leaving a remainder.

3. Can this calculator use negative numbers?

Yes. The division chain uses absolute values. The Bézout identity then adjusts coefficient signs according to the original inputs.

4. What happens if one input is zero?

If one value is zero, the GCD is the absolute value of the other value. If both values are zero, the GCD is undefined.

5. What are Bézout coefficients?

They are integers x and y that satisfy ax + by = gcd(a, b). They are useful in modular arithmetic and number theory proofs.

6. When are two numbers coprime?

Two numbers are coprime when their GCD is 1. This means they share no positive divisor other than 1.

7. What is the LCM formula?

The least common multiple is found by dividing the absolute product of both numbers by their GCD. The formula is |a × b| ÷ GCD.

8. Why export results?

Exporting saves the result, division steps, coefficients, and notes. This is useful for assignments, reports, audits, and later review.