How many positive integers $x \le 3600$ are there such that $\gcd(3600, x)=9$?

Question

I'm trying to answer this question which has a hint: think about $\mathbb Z_{3600}$.

I tried to set up a linear equation,$\mod{3600},$ without any success. Not even the factorization of $3600$ gives me any ideas on how to set the problem.

Any help?

Answer 1

Note that$$\gcd(3600,x)=9\Longrightarrow x=9y\quad,\quad y\in \Bbb Z_{400}\to \gcd(400,y)=1$$therefore the number of solutions of $\gcd(3600,x)=9$ on $\Bbb Z_{3600}$ is equal to number of solution of $\gcd(400,y)=1$ on $\Bbb Z_{400}$ which is $\phi(400)$.

Answer 2

A (different) hint: notice that

$$ \gcd(3600,x) = 9 \iff \gcd(400,x/9) = 1.$$

Answer 3

Hint: Such a number must be of the form $9k$, where $k\le 400$ and $\gcd(400,k)=1$.