Moduloissa

Moduloissa is a Finnish term used in mathematics to refer to calculations within a modular system defined by a modulus n. In modular arithmetic, a and b are congruent modulo n if n divides a−b; this is written a ≡ b (mod n). The residue class modulo n can be represented by integers 0 through n−1, and every integer is congruent to exactly one of these residues. The set of all residues modulo n, with addition and multiplication performed and then reduced modulo n, forms the ring Z/nZ; when n is prime, Z/nZ is a field, enabling division by nonzero residues.

Common operations are performed modulo n: addition, subtraction, and multiplication are closed, exponents can be reduced via Euler's or Fermat's theorems, and solving congruences a ≡ b (mod n) means finding all a that share the same residue as b modulo n. The Chinese remainder theorem gives a way to combine congruences modulo pairwise coprime moduli.

Examples: 23 ≡ 5 (mod 6) because 23−5=18 is divisible by 6; 7+11 ≡ 18 ≡ 0 (mod 6); 4^3 ≡ 64 ≡ 4 (mod 10).

Applications include number theory, cryptography (RSA, elliptic curves rely on arithmetic modulo large primes), calendar calculations (days of the week cycle every 7), hashing, and randomized algorithms.

Etymology: the term derives from the mathematical Latin modulus and its Finnish adaptation.