Multiple private keys with RSA Say I want generate pair of public/private keys. First of all lets choose two prime numbers $q = 17, p = 11$. Now, lets choose public key $e=7$ and try to determine the matching private key $d$:
$$d\cdot e \mod (q-1)(p-1) = 1$$
$$d\cdot 7 \mod 16\cdot 10 = 1$$
$$7d \mod 160 = 1$$  

So, there are many solutions for this equation, such $23$, $183$, $343$ and in general $d = 160k+23$ for every $k\in Z$.

But RSA should have single private key for public key, so why can I get many solutions/private keys? I tested them, and they can decrypt the encryption of the public key $e$.

I know that $q$ and $p$ should be destroyed and without them it is hard to calculate the private key. I just don't understand why there are many private keys for one public key...
Well,


You're working in the multiplicative group. 
There is a theorem (which is proven by group axioms), that the inverse element is unique ($d$ is inverse to $e$).
When you say $d = 160 \cdot k + 23$ for every $k$ from $Z$, you actually should mean that in the context mod $160$ then your $d \cdot k$ for any $k$ is actually the same element.


I think the mindtrick here is that:

You claim: $d = 160 \cdot k + 23$ for every $k$ in Z; oh, there is a lot of such elements!
NO. You're in the multiplicative group. You can write this element different ways. But it is still the same element.

You do not work with "numbers". You work with an element of multiplicative group. Which can be represented many ways (see your claim about $k$). But it's the same element. In the context of 'multiplicative group mod $160$, elements $23$ and $183$ are the same elements. 

You can easily prove that $23 \bmod 160 = 183 \bmod 160$. So $183$ equals $23$ in group mod $160$. That means you can write on sheet of paper $183$ or $23$ or whatever $160 \cdot k + 23$ you like, and you will be talking about the same element, which is unique.

The key is therefore also unique - your key is an element of a multiplicative group, which can be written in many ways ($160 \cdot k + 23$, where $k$ is in $Z$)