Euler Theorem. (4)

     If n = p₁^a1 p₂^a2 ... p_k^ak, then

j(n) = n (1 - 1/p₁) (1 - 1/p₂) ... (1 - 1/p_k) .

     Proof. We have:

j(n) = j(p₁^a1) j(p₂^a2) ... j(p_k^ak) =

(p₁^a1 - p₁^a1^-1) (p₂^a2 - p₂^a2^-1) ... (p_k^ak - p_k^ak^-1) =

p₁^a1 p₂^a2 ... p_k^ak (1 - 1/p₁ ) (1 - 1/p₂) ... (1 - 1/p_k),

what coincides with equality that we are proving, what was to be proved.