Questions tagged [totient-function]

Questions on the totient function $\phi(n)$ (sometimes $\varphi(n)$) of Euler, the function that counts the number of positive integers relatively prime to and less than or equal to $n$.

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Finding missing integers from multiplicative group mod 56

From Gallian's Contemporary Abstract Algebra (5ed), end of Chapter 2: The integers 5 and 15 are among a collection of 12 integers that form a group under multiplication mod 56. List all 12. I'm having ...
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Prove that the set $A(d)=\{m\in\mathbb{Z}|1\leq m\leq n\quad\&\quad gcd(m,n)=d\}$ are pairwise disjoint

Let $n\in\mathbb{Z}_+$ then $n=\sum_{d|n} \phi(d)$, where $\sum_{d|n}$ denotes the sum over all of the divisors of $n$ and $\phi(d)$ is the Euler phi function which gives the number of integers less ...
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Find all $m,n$ such that $\varphi(mn) = \varphi(n)$. [duplicate]

Find all $m,n$ such that $\varphi(mn) = \varphi(n)$. I applied cases for the question and got $2$ cases. Case: $\gcd(m,n) = 1$ Case : $\gcd(m,n) = d$, where $d$ is any arbitrary integer So in case-1 ...
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How does the following corollary 3 follow from corollary 2 in I.N Herstein?

In I.N Herstein's Topics in Algebra is given certain corollaries which follow from a definition. Quoting: COROLLARY 2 $\ \ \ \ \ \ \ \ \$If G is a finite group and $a \in G$, then $a^{o(G)}=e$ This ...
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Relation of totient, cycle length of multiplication and divisors

I am reading about the dynamics of multiplication $\pmod m$ , and the totient function. So from what I have understood, the totient function $\phi(n)$ is the cardinality of $\Phi(n)$ which is the set ...
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If n is even, prove that the summation (indexed over the divisors of n) ϕ(d)µ(d) = 0 [duplicate]

I am having great difficulty with the following proof: Prove that if $n$ is even, $\sum_{d|n} μ(d)ϕ(d) = 0$ First, I noticed a general pattern that we will use later: For any integer $a$, we see that ...
Is there any unique value for Euler Phi Function $\phi(n)$?
I know $\phi(n) = \phi(2n)$ for $n$ odd and greater than $1$. I wonder if there any value $k$ such that $\phi(n) = k$ for a unique $n$. $\phi(2) = \phi(1) = 1$ so of course $1$ cannot be that value. I ...