Hint: compute $d$ for the alternate value of $e$ considered. Isn't something rather remarkable? What is the probability that it occurs by chance? 

While that is totally immaterial given that $p$ and $q$ are billions-n'-billions times too small to prevent factorization, what if we scaled things? 

Facts relevant to this (and many other sensible RSA questions) can be found by combing Dan Boneh's classic [_Twenty Years of Attacks on the RSA Cryptosystem_][1] (in [Notices of the AMS, Vol. 46, No. 2][2], pp. 203-213, 1999).<br>
<sub>I would have properly cited Jingjing Wang's [followup attempt][3] if it had acknowledged what inspired it</sub>.


  [1]: https://www.ams.org/notices/199902/boneh.pdf
  [2]: https://www.ams.org/notices/199902/199902-body-ps.html
  [3]: https://cryptjwang.files.wordpress.com/2012/05/rsa_attacks.pdf