Exercise 844Number theory › CongruencesL1Added 3 Nov 2026

Euler's theorem

#Euler theorem#totient#modular exponentiation

Statement

State Euler's theorem and compute $3^{2024}\pmod{10}$ .

Hint

For $\gcd(a,n)=1$ : $a^{\varphi(n)}\equiv1\pmod n$ . Here $\varphi(10)=4$ .

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Theorem. If $\gcd(a,n)=1$ , then $a^{\varphi(n)}\equiv1\pmod n$ .

Parameters. $n=10$ , $\varphi(10)=4$ , $a=3$ coprime to $10$ .

Compute. $2024=4\cdot506$ , so $3^{2024}=(3^4)^{506}\equiv1^{506}=1\pmod{10}$ .

Conclusion. $3^{2024}\equiv1\pmod{10}$ .

Check. $3^4=81\equiv1\pmod{10}$ . Correct.