Verified answer

Case 1: 3 does not divide n.

Then, φ(3n) = φ(3) φ(n) = (3 - 1) φ(n) = 2 * φ(n).

Case 2: 3 divides n.

Writing n = 3^k * q, where q does not divide 3:

φ(n)

= φ(3^k * q)

= φ(3^k) φ(q)

= 3^k (1 - 1/3) φ(q)

= 2 * 3^(k-1) * q

φ(3n)

= φ(3^(k+1) q)

= φ(3^(k+1)) φ(q)

= 3^(k+1) (1 - 1/3) φ(q)

= 2 * 3^k * q

= 3 * (2 * 3^(k-1) * q)

= 3 * φ(n).

-------------

I hope this helps!