A number-theoretic result: for integers a and n with gcd(a,n)=1, a^{φ(n)} ≡ 1 (mod n), where φ(n) is Euler's totient function giving the order of the unit group (Z/nZ)× when counted multiplicatively.

Principle

The multiplicative group of units modulo n has order φ(n), so by Lagrange's theorem any unit raised to the group's order equals the identity modulo n.

Demonstration

Take a=3, n=10: φ(10)=4 and 3^4=81≡1 (mod 10). More generally, compute powers modulo n for any unit and reduce exponents modulo φ(n) when appropriate.

Misapplication

Using the congruence when gcd(a,n)≠1 (for example a and n not coprime) or assuming φ(n) is the minimal exponent for all units; the true minimal universal exponent may be the Carmichael λ(n).

Consequence

Provides exponent reduction rules in modular arithmetic and underlies correctness and key steps in cryptographic protocols and algorithms that rely on modular exponentiation.

Reversal

Fermat's little theorem is the special case n prime; conversely, knowing the congruence for all a coprime to n imposes structural constraints on the multiplicative group of (Z/nZ)×.

Boundary

Requires a and n to be integers with gcd(a,n)=1; it does not assert the smallest exponent that sends every unit to 1 (that is governed by λ(n)) and does not extend to zero divisors modulo n.

Semantic Tension

Often conflated with Fermat's little theorem or with statements about orders of individual elements; Euler's theorem is a group-order statement, not a claim about primitive roots or minimal exponents for all units.

Synthesis

Euler's theorem packages the group-theoretic fact that units modulo n have finite order φ(n) into a concrete modular congruence a^{φ(n)}≡1, enabling exponent reductions and connections between multiplicative structure and modular arithmetic.