Anyone know about RSA Algorithm? If you don't, I will explain it. So to get a public and private key in RSA we need two primes number, let $p$ and $q$ be two prime number then we count $n=p.q$ and $\varphi (n)=(p-1)(q-1)$, next we choose public key e which prime relative with $\varphi (n)$, that is $gcd(e,\varphi(n))=1$, and $1<e<\varphi (n)$, next we find private key d which satisfied $e.d \equiv 1\pmod{\varphi (n)}$, example we have $p=7, q=11$ so that $\varphi (n)=60$, then we choose $e=13$, and finally we find $d=37$ and can't be another number.
Question: The question is for every different $e$ then $i$ get different $d$ or the value of is unique, is that true? If yes how can $i$ prove it mathematically? Let say $i$ want to prove for every different value of e, the value of d is also different or d is unique? The idea that $i$ have is using contraposition let $e_1.d=e_2.d$ then $e_1=e_2$, anyone can help me to prove that? If you have another method you can use it as long as it true. Thank you.
