Find all values of $n$ such that $\phi(n)=100$

Question

How do I go about solving this problem, and similar ones where I have to find the number of values that satisfy this phi function?

Thanks!

Answer 1

Here's the general approach.

1. Write $n = \prod p_i^{k_i}$ in its prime factorization.

2. Then $\phi(n) = \prod\phi(p_i^{k_i}) = \prod p_i^{k_i - 1} (p_i - 1)$, using the property that $\phi$ is multiplicative and computing directly its evaluation on any prime power.

From there, you can logic it all out!

One quick check is to see if there are trivial answers: if we are solving $\phi(n) = m$ and $m+1$ is prime, then $n = m+1$ is a solution.

I'll show how a couple observations about $\phi$ make the analysis simple.

Firstly, notice each odd prime in the factorization of $n$ contributes at least $1$ factor of $2$ to $\phi(n)$. Since $100 = 4 * 25$, this is a powerful restriction saying the prime factorization contains at most $2$ distinct odd primes.

Secondly we can see that the presence of $2^m$ in the factorization of $n$ contributes a factor of $2^{m-1}$ to $\phi(n)$, and in particular that $\phi(2n) = \phi(n)$ for $n$ odd.

These two conditions already restrict the factorization of any $n$ such that $\phi(n) = 100$ as follows (with $p_k >2)$:

1. $n = 2^j*p^i$ (with $j \leq 2$)
2. $n = 2^j*p_1^i * p_2^j$ (with $j \leq 1$)

Note that $2$ is not satisfiable. Ignoring the factor of $2$ in $n$ (which is inconsequential to $\psi(n)$) this would force $p_1^{i-1}p_2^{j-1} (p_1-1)(p_2-1) = 2*2*5*5$. Remember that $p_1-1$ and $p_2-1$ are even, distinct, and $p_k \geq 2$, but that $p_1 - 1 \in \{2,4,10,20,50,100\}$ necessarily. What possibilities for $p_1$ and $p_2$ does this leave you with? You can proceed to show that none of them are valid.

Now we just have to analyze $1$.

Again you can look at all the ways to form $p-1$ from even groupings of factors of $100$, and then reason which possibilities of $p$ are possible. There aren't very many cases to look at.

The important thing is that you can always restrict the form of $n$ with basic properties of $\phi$ and then do case analysis. For larger equations $\phi(n) = m$ with $m$ big, you might want to look into studying properties of $\phi$ in more detail, perhaps even specific to the factorization of $m$.