Exercise 2109Number theory › CongruencesM1Added 3 Nov 2026

The Carmichael function $\lambda(n)$

#Carmichael function#lcm#exponent

Statement

Define $\lambda(n)$ and compute it for $n=15$ .

Hint

$\lambda(n)$ is the lcm of the orders of $(\mathbb{Z}/n\mathbb{Z})^*$ . For $p^k$ , $\lambda(p^k)=\varphi(p^k)$ for odd $p$ .

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Solution

Definition. $\lambda(n)=\operatorname{lcm}\{\operatorname{ord}_n(a):\gcd(a,n)=1\}$ . It is the smallest $m$ with $a^m\equiv1\pmod n$ for all coprime $a$ .

For $n=15=3\cdot5$ : $\lambda(15)=\operatorname{lcm}(\lambda(3),\lambda(5))=\operatorname{lcm}(2,4)=4$ .

Verify. $2^4=16\equiv1\pmod{15}$ ; $4^2=16\equiv1\pmod{15}$ ; $7^4=2401\equiv1\pmod{15}$ ; etc.

Conclusion. $\lambda(15)=4$ , which divides $\varphi(15)=8$ .

The Carmichael function $\lambda(n)$

Statement

Solution

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