Modular Arithmetic Calculator – Mod Operations

Modular arithmetic wraps results around a modulus m, keeping only remainders. Adding 7 + 9 hours on a 12-hour clock gives 16 mod 12 = 4 — 4 o’clock. Multiplication wraps the same way: 5×5 = 25 mod 12 = 1, so 5 is its own inverse modulo 12. The calculator supports +, −, × and powers mod m, reducing after every operation so intermediate values stay small — computing 3⁴ mod 5 as successive reductions (3² = 9 ≡ 4, then 4² = 16 ≡ 1) rather than 81 first. Congruence testing checks whether m divides the difference: 17 ≡ 5 (mod 12) because 12 | 12. Negative operands normalize into [0, m), so −7 mod 12 = 5. This system powers RSA encryption (exponentiation mod huge semiprimes), hash functions, cyclic schedules, and music theory’s 12 semitone clock.

1. 1

   Choose the modulus m

   12 for clock math, 7 for weekdays, 2 for parity

2. 2

   Enter operands and operator

   a and b with +, −, ×, or power

3. 3

   Reduce at every step

   Intermediate results are wrapped mod m immediately

4. 4

   Read the congruence

   The a ≡ r (mod m) statement is displayed

• 1

  Reduce early, reduce often — values never need to exceed m²

• 2

  a ≡ b (mod m) means m divides the difference a−b

• 3

  Inverses exist only when gcd(a, m) = 1

• Reducing only the final result and overflowing intermediate powers

• Assuming subtraction never goes negative — wrap with +m

• Dividing in modular arithmetic as if rationals were allowed

• Expecting an inverse for non-coprime pairs: 4 has none mod 8