Number Theory Calculator

How to use the calculator

1. Select a mode and enter the values shown for it.
2. In Modular arithmetic, choose an operation from Operation. Enter b when the exponent field appears.
3. In Permutations & combinations, select Without repetition or With repetition.
4. Select Calculate to display the exact result and generated calculation details.
5. Select Copy result or Download result (.txt) to keep the result as plain text. Select Reset to clear the fields and hide the current result.

Mode-specific inputs and controls

• Greatest common factor and Least common multiple accept two or more comma- or space-separated integers. Each list accepts at most 50 integers. GCF accepts zero if at least one value is nonzero; LCM rejects any list containing zero. Negative signs are ignored for these calculations.
• Prime factorization accepts one integer with up to 15 digits. Zero has no prime factorization. For negative input, the calculator factors the absolute value and reports the sign separately.
• Modular arithmetic uses Operation for a mod n, additive inverse, multiplicative inverse, or modular exponentiation. The modulus n cannot be zero, and exponent b must be nonnegative when shown. A multiplicative inverse exists only when GCF(a, n) = 1.
• Factorial accepts a nonnegative integer from 0 to 10,000. It returns 0! = 1 for zero.
• Permutations & combinations accepts nonnegative n and r values from 0 to 10,000 and returns both results for the selected repetition setting. Without repetition, r > n gives 0 for both.

Calculation details and limitations

The result includes formulas and generated calculation details, but it does not list every underlying trial divisor or operation.

• Factorial multiplication is expanded through 15 terms. For larger inputs, the chain is abbreviated; the result and digit count are still shown.
• Modular exponentiation uses repeated squaring and reports the number of squaring steps rather than every multiplication.
• Prime factorization records the prime factors found by trial division, not every divisor tested.
• GCF, LCM, and modular arithmetic accept integers up to 100 digits. Factorials, permutations, and combinations use the 10,000 limit stated above.
• For negative modular inputs, a mod n returns a result with the sign of n or zero. Additive inverse is normalized using the absolute value of the modulus.

Privacy and calculator processing

The arithmetic is implemented in JavaScript on this page. The local implementation does not document broader analytics or telemetry handling, so this page makes no claim about whether other page information is sent to a server.

Frequently asked questions