modulo97

Modulo97 refers to arithmetic conducted with the modulus 97. The integers modulo 97 form a finite ring Z/97Z, and since 97 is a prime number, this ring is a field. In this system, two integers a and b are congruent modulo 97 if their difference is divisible by 97, written a ≡ b (mod 97). Each integer has a unique residue class, and every residue class can be represented by a number in the range 0 to 96.

Operations are performed modulo 97: add, subtract, and multiply, then reduce the result modulo 97. Division is defined only when the divisor is invertible modulo 97; because 97 is prime, every nonzero residue has a multiplicative inverse. For example, 5 has inverse 39 because 5×39 = 195 ≡ 1 (mod 97). The set of nonzero residues modulo 97 forms a multiplicative group of order 96 and is cyclic; there exist primitive roots, and there are φ(96) = 32 of them.

Common uses include solving congruences, studying quadratic residues, and constructing finite fields for theoretical purposes. While modulo 97 is far too small for secure cryptographic schemes, the concepts generalize to larger prime moduli and to composite moduli in different algebraic structures. In practice, modulo 97 is often employed as a didactic example to illustrate modular arithmetic, residues, inverses, and modular equations.