algorithm – how to derive from private key to public key in math

I am gonna write down formulas that I know and use to generate RSA keys.

  1. we choose p, q
  2. N = p*q
  3. f(n) = (p-1)(q-1)
  4. choose e such as
  • 1 < e < f(n)
  • coprime with N, f(n)
  1. choose d so that d * e (mod f(n) ) = 1

That’s it. With these, if we have p=2 and q=7, I succesfully get d=11 and e=5 which is correct.

Now Imagine, that I only have private key which is (11,14) (d=11, N=14). How do I get e=5. I understand that with d and N, you can’t directly get e, but as RSA works, it tries different variants of e , then checks and if it’s valid, that’s how you get public key from private key.

Can anyone explain to me what steps should I take here to figure out what e could be and then from those, which e should I choose ?