Euler Phi Function is not Completely Multiplicative

Theorem

The Euler $\phi$ function is not a completely multiplicative function.

That is, it is not always the case that:

$\map \phi {m n} = \map \phi m \map \phi n$

where $m, n \in \Z_{>0}$.

Proof

Proof by Counterexample

\(\ds \map \phi 6\)

\(=\)

\(\ds 2\)

$\phi$ of $6$

\(\ds \map \phi {10}\)

\(=\)

\(\ds 4\)

$\phi$ of $10$

\(\ds \map \phi {60}\)

\(=\)

\(\ds 16\)

$\phi$ of $60$

Hence we see:

$6 \times 10 = 60$

byt:

$\map \phi 6 \times \map \phi {10} \ne \map \phi {60}$

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Properties of the Natural Numbers: $\S 25$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.3$: Congruences
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $30$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): arithmetic function