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...