Modulo Calculator

Modulo arithmetic wraps numbers around like a clock: 16 hours after nine o'clock is not 25 o'clock but 1 o'clock, because several full loops of twelve are stripped away. The operation that strips them is the modulus: a mod b is the residue left after removing as many full b-sized chunks as possible. It underlies checksums, calendars, hashing, cryptography and endless number-theory questions - and it must be exact, because one wrong digit is a completely different residue.

Modulo Calculator computes a mod b with BigInt precision for any integers, positive or negative, returning the Euclidean remainder and the full identity a = q x b + r, plus the truncated quotient-and-remainder pair. A second mode computes a^e mod m by fast modular exponentiation. Everything runs locally in the browser.

• BigInt exactness: Hundreds of digits with zero rounding - remainder, quotient and power all precise.
• Euclidean remainders: Negative inputs resolve to the 0 to |b| - 1 residue with the identity shown.
• Floor and truncated views: Both quotient conventions visible, so the math lines up with any book.
• Modular exponentiation: a^e mod m by repeated squaring - astronomically big exponents, fast.
• Divisibility badge: When b divides a, the result 0 is flagged immediately.
• Zero-divisor guard: Dividing by zero is caught with a clear message instead of a crash.
• Copy the result: One click puts the full identity on the clipboard.
• Reference tab: Clock arithmetic, negative residues, the exponentiation algorithm.
• Free and private: Everything runs in the browser; nothing is uploaded or stored.

1. Enter a and b - any integers, negative signs allowed, any length.
2. Read the remainder - the Euclidean a mod b result, big and bold.
3. Check the identity - a = q x b + r with the exact floor quotient.
4. Switch to power mode - a^e mod m with any exponent.
5. Copy the full result row for your records.

Example 1 - Clock arithmetic. a = 25, b = 12: the tool returns 1, the hour after two full twelve-hour loops - 25 = 2 x 12 + 1.

Example 2 - Negative dividend. a = -17, b = 5: remainder 3 with the identity -17 = -4 x 5 + 3 - the Euclidean residue never goes negative.

Example 3 - Negative divisor. a = 17, b = -5: remainder 2, quotient -3: 17 = -3 x -5 + 2, and the remainder still sits between 0 and |b|.

Example 4 - Divisibility. a = 4032, b = 63: remainder 0 - 63 divides 4032 exactly, and the badge announces it.

Example 5 - Huge torsion. a = 2 to the 1000, b = 97: the giant power mod 97 reduces to a tiny residue by repeated squaring - seconds, not years.

• Absolute precision: BigInt arithmetic means the residue is right for any digit count.
• Convention-consistent: Euclidean remainders never surprise a negative input.
• Two views, one truth: Floor and truncated quotients explained, identity verified.
• Power mode included: The same page handles discrete exponentiation without a second tool.
• Graphical clarity: Residue, quotient, divisibility badge and identity in one glance.
• Private and free: Nothing is uploaded, stored or logged.