Recently, I have found this problem:
Determine for which values of $\eta\in\mathbf{N}$ and $\varphi\in\mathbf{N}_0$, the quantity: $$\gamma=\frac{10^2\cdot\eta}{\varphi\cdot\eta-1}$$ is a positive integr ($\gamma\in\mathbf{N}$).
To solve this problem, I began studying the case where $\gamma=0$. Here $\varphi$ can be every number and $\eta=0$.
After that, I passed to the case where $\varphi=1$. So, I have: $$\gamma=\frac{10^2\cdot\eta}{\eta-1}$$ The tecnique I used, is trying to transform the numerator into the denominator, so: $$\gamma=\frac{10^2\cdot\eta-10^2+10^2}{\eta-1}=\frac{10^2\cdot(\eta-1)+10^2}{\eta-1}=10^2+\frac{10^2}{\eta-1}$$ Now, $10^2$ is an integer, so also $\frac{10^2}{\eta-1}$ must be an integer. In conclusion if $\varphi=1$ then the solutions are: $\eta=\{2,3,5,6,11,21,26,51,101\}$.
Then, I studied the case when $\varphi\mid 10^2$. Here, I can use the tecnique shown above. In fact: $$\gamma=\frac{10^2\cdot\eta-\frac{10^2}{\varphi}+\frac{10^2}{\varphi}}{\varphi\cdot\eta-1}=\frac{\frac{10^2\cdot\varphi\cdot\eta-10^2\cdot\varphi}{\varphi}+\frac{10^2}{\varphi}}{\varphi\cdot\eta-1}=\frac{10^2}{\varphi}+\frac{10^2}{\varphi\cdot(\varphi\cdot\eta-1)}$$ In the hypotesis written abvove $\frac{10^2}{\varphi}$ is an integer, so $\frac{10^2}{\varphi\cdot(\varphi\cdot\eta-1)}$ must be an integer for a fixed $\varphi$.
The case where $\varphi\nmid10^2$, I think, is more complicated.
So, can we build a general method to find $\eta$ when $\varphi$ varies over the positive integers?
$\eta$ must be a factor of $\gamma$ (because $\phi \eta - 1$ and $\eta$ are relatively prime). Then $\gamma$ is a positive integer iff $\eta \phi - 1$ is a positive factor of 100. Clearly, there are only finitely many such combinations.