How to choose the appropriate public (i, m) and private (j, m) keys? I studied some encryption and decryption and I have found some very interesting problem to solve on the internet. I hope I am writing to right site - there are so many in StackExchange otherwise I hope some moderator will move it.

So, how to choose the appropriate public lets say (i, m) and private (j, m) keys which could be subsequently used to (de)encrypt in RSA when there are primes lets say p = 17 and q = 5?

I tried something like:

17*5=85

(17-1)*(5-1)=64


64 should be magic number but still do not know what to do next, can anyone help me here?
 I would really appreciate how to choose and in which way these keys ...
I assume that $m = pq$ (85) is the modulus, and the number you computed (64) is $\phi(m)$. Now we must choose the exponents ($i$ for public, $j$ for secret? , weird notation) in such a way that $ij = 1$ modulo $\phi(m)$. In fact we can work modulo 16 if we like, because that is the least common multiple of $p-1$ and $q-1$. So if we choose $i = 3$, we choose $j=11$, as $33 \equiv 1 \mod 16$. Or $i=11$, $j=3$, this is symmetrical. Or we could choose $i = 5$ and then $j=13$ works, or vice versa again.

Then encryption of a number smaller than $m$ is exponentiation to the power $i$, modulo $m$ and decryption is exponentiation to the power $j$, modulo $m$.

Of course these numbers are way too small to provide any security, but that's the general idea.