Concepts5

Modular Arithmetic Pitfalls

Modular arithmetic is about working with remainders, but programming languages often return negative remainders, so always normalize with (a % MOD + MOD) % MOD.

#modular arithmetic#modular inverse#fermats little theorem+12

∑MathIntermediate

Lucas' Theorem

Lucas' Theorem lets you compute C(n, k) modulo a prime p by working digit-by-digit in base p.

#lucas theorem#binomial coefficient modulo p#prime power modulus+12

∑MathIntermediate

Euler's Theorem

Euler’s Theorem says that if a and n are coprime, then a raised to the power φ(n) is congruent to 1 modulo n.

#euler totient#euler theorem#modular exponentiation+12

∑MathIntermediate

Chinese Remainder Theorem

The Chinese Remainder Theorem (CRT) reconstructs an integer from its remainders modulo pairwise coprime moduli and guarantees a unique answer modulo the product.

#chinese remainder theorem#crt#modular arithmetic+12

∑MathIntermediate

Extended Euclidean Algorithm

The Extended Euclidean Algorithm finds integers x and y such that ax + by = gcd(a, b) while also computing gcd(a, b).

#extended euclidean algorithm#bezout coefficients#gcd+12