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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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

My bad, I misread what you wrote. --RDBury (talk) 16:55, 30 November 2015 (UTC)[reply]