Wikipedia:Reference desk/Archives/Mathematics/2015 November 30

= November 30 =

Euler's totient function
For two positive integers m and n, when is $$\phi(mn) = m\phi(n)$$ true? GeoffreyT2000 (talk) 03:54, 30 November 2015 (UTC)
 * When the primes dividing m all divide n. For example ϕ(100⋅40)=1600=100ϕ(40). Note, it's always the case that ϕ(mn)≤mϕ(n). --RDBury (talk) 07:00, 30 November 2015 (UTC)
 * Coprimality depends only on the distinct prime factors. If every prime dividing m also divides n, then mn has the same distinct prime factors as n and hence an integer is coprime to mn if and only if it is coprime to n. GeoffreyT2000 (talk) 14:48, 30 November 2015 (UTC)
 * That's not what I meant. For example every prime dividing 4 (namely 2) also divides 6. In this case ϕ(4⋅6)=4ϕ(6) but you can't say an integer is coprime to 4 iff it's coprime to 6 (3 being a counterexample). --RDBury (talk) 15:13, 30 November 2015 (UTC)
 * But it is true that an integer is coprime to 24 if and only if it is coprime to 6. I said "coprime to mn", not "coprime to m". GeoffreyT2000 (talk) 15:29, 30 November 2015 (UTC)
 * My bad, I misread what you wrote. --RDBury (talk) 16:55, 30 November 2015 (UTC)