residuesalso

Residuesalso is a term used in discussions of modular arithmetic to describe the joint residue information of an integer with respect to a chosen finite set of moduli. In informal use, it refers to the multi-modulus residue signature of a number, capturing its remainders under several moduli simultaneously. While not a standard term in most textbooks, it can be a helpful shorthand in explanations and exercises.

Formally, let M = {m1, m2, ..., mk} be a finite set of positive integers. For an integer x, the residuesalso with respect to M is the k-tuple (r1, r2, ..., rk) where ri = x mod mi. Two integers a and b have the same residuesalso with respect to M if and only if a ≡ b (mod mi) for every i. Consequently, a and b are congruent modulo L, where L = lcm(m1, m2, ..., mk). Thus there are L distinct residuesalso classes, corresponding to residue classes modulo the least common multiple of the moduli.

Examples: with M = {3, 5, 7} and x = 23, the residuesalso is (2, 3, 2). Any y with the same tuple satisfies y ≡ 23 (mod 3), (mod 5), and (mod 7), hence y ≡ 23 (mod 105).

Relationships and uses: residuestoes are closely tied to the Chinese Remainder Theorem, which guarantees a unique solution modulo L for compatible systems of congruences. In pedagogy, residuesalso helps illustrate how information about several moduli encodes a number, and in cryptographic or data-encoding contexts, it can serve as a compact modular signature. Limitations include that the construction offers no new algebraic structure beyond what is already captured by congruence modulo L.

See also: modular arithmetic, residue class, Chinese Remainder Theorem, least common multiple.