5. Let *n* = *p*₁^*k*₁ *p*₂^*k*₂ ... *p*_*m*^*k*_*m* be the unique prime factorization of *n*. Define λ(*n*) as the least common multiple (see p. 259) of all φ(*p*_*i*^*k*_*i*), where *i* = 1, 2, ..., *m*. Prove:

(a) λ(*n*) | φ(*n*);

(b) *a*^λ(*n*) = 1 mod *n* for all [*a*]_*n* ∈ Z_*n*;

(c) if *n* is a Carmichael number, then *k*_*i* = 1 for all *i* = 1, 2, ..., *m*;

(d) if *n* is a Carmichael number, then *n* is the product of at least three different primes.

(e) Is 27935017 a Carmichael number?