Euler’s Theorem Calculator
This calculator allows you to apply Euler’s Theorem to compute modular exponentiation when the base and modulus are coprime. It’s widely used in number theory and cryptography, especially in RSA and modular arithmetic problems.
Compute Modular Powers Using Euler’s Theorem
Euler’s Theorem Formula
$$\text{If } \gcd(a, n) = 1,\ \text{then:} \quad a^{\phi(n)} \equiv 1 \pmod{n}$$
Explanation:
If $$a$$ and $$n$$ are coprime (i.e., their greatest common divisor is 1), then raising $$a$$ to the power of $$\phi(n)$$ (Euler’s totient function of n) will result in 1 modulo $$n$$. This property allows us to reduce large exponents in modular calculations.
Euler’s Theorem – Calculation Example
Let a = 3, n = 10
- Check gcd(3, 10) = 1 → OK
- φ(10) = 4
- 3^4 mod 10 = 81 mod 10 = 1
Result: 3⁴ ≡ 1 mod 10 → Euler’s Theorem holds
Euler’s Theorem generalizes Fermat’s Little Theorem and is crucial in modular arithmetic. It helps simplify calculations with large exponents and is a foundation for RSA encryption. This calculator verifies the theorem or helps with efficient power reduction modulo n.