How to Apply Euler’s Theorem and Fermat’s Little Theorem Definition, Types, Examples & Uses

How to Apply Euler’s Theorem and Fermat’s Little Theorem

Subject: Mathematics

Book: Maths Mastery

Fermat’s Little Theorem says that if p is prime and gcd(a,p)=1, then a^(p–1)≡1 (mod p). Euler’s theorem generalizes it, stating a^φ(n)≡1 (mod n) for gcd(a,n)=1. These reduce exponents in modular arithmetic. For example, to find 3^100 mod 11, note φ(11)=10, so 3^100 = (3^10)^(10) ≡1^(10)≡1 mod 11. Such exponentiation shortcuts appear in coding, cryptography (like RSA), or advanced number theory tasks. Familiarity with these theorems speeds up computations involving large powers mod n.