<s>[INST] For which values of $\eta\in\mathbf{N}$, $\frac{10^2\cdot\eta}{\varphi\cdot\eta-1}$ is an integer ($\varphi\in\mathbf{N}_0$)? 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?
 [/INST] Observe that $\varphi\cdot\eta -1$ is prime with $\eta$ (use Euclidean Division). So $\gamma$ is an integer iff $\varphi\cdot\eta -1$ divides $10^2$, do you know how to proceed?
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