True or false: "$\varphi(12 \cdot 15) = \varphi(12) \cdot \varphi(15)$"

False — Euler's phi function is only multiplicative when the arguments are coprime: $\gcd(12, 15) = 3 \neq 1$, so the product rule does NOT apply.

The rule: $\varphi(m \cdot n) = \varphi(m) \cdot \varphi(n)$ only when $\gcd(m, n) = 1$

Checking this case:

• $\gcd(12, 15) = 3 \neq 1$ → rule does NOT apply
• $\varphi(12) \cdot \varphi(15) = 4 \cdot 8 = 32$ ← wrong answer
• Correct: $\varphi(180) = \varphi(2^2 \cdot 3^2 \cdot 5) = 180 \cdot \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{4}{5} = 48$

Contrast with valid applications:

• $\varphi(7 \cdot 11) = \varphi(7) \cdot \varphi(11) = 6 \cdot 10 = 60$ ✓ (since $\gcd(7, 11) = 1$)
• $\varphi(4 \cdot 9 \cdot 5) = \varphi(4) \cdot \varphi(9) \cdot \varphi(5) = 2 \cdot 6 \cdot 4 = 48$ ✓ (all pairwise coprime)

This is a classic pitfall. Always check $\gcd$ before applying the multiplicative property. When factors share a common prime, use the prime factorization formula instead: $\varphi(n) = n \cdot \prod_{p|n}(1 - 1/p)$.

Go deeper:

• Euler's totient function (Wikipedia) — why multiplicativity needs coprime arguments.