Use-Case of Euler Totient Function in Competitive Programming
Calculating Large Exponential Value % mod:
Euler’s Totient Theorem states that if a and n are coprime (gcd(a, n) = 1), then [Tex]a^{φ(n)} ≡ 1 (mod n) [/Tex]. This theorem is used to find the remainder of a number raised to a large power modulo n efficiently. Competitive programming problems often require solving such congruence relations.
Calculating Co-Prime Pairs:
You may encounter problems that require counting the number of pairs (a, b) such that 1 ≤ a, b ≤ n and gcd(a, b) = 1. The Euler Totient function can be used here to calculate [Tex]\phi(n) [/Tex] and then determine the count of coprime pairs.
Eulter Totient Function as DP-state:
In dynamic programming (DP) problems, you might use [Tex]\phi(n) [/Tex] as a state in your DP table when solving combinatorial or counting problems involving modular arithmetic.