Remark

The Euler indicator of a number can be expressed in terms of its prime divisors as follows: if n = p₁^a₁ ··· p_t^a_t, then

(n) = p₁^a₁-1 (p₁-1) ··· p_t^a_t-1(p_t-1).

In fact this is not hard to prove using the lemma and the following consequence of the Chinese remainder theorem.

(pq) = (p) (q)

whenever gcd(p,q) = 1.