RSA Key generation Public exponent too big I'm currently writing a paper about RSA (a self-chosen subject).
I'm writing about the key-generation in RSA, and I have problem finding the public exponent e.

I have chosen p = 61 and q = 53. 

Then I determined my modulus n = p ⋅ q = 3233 

and my phi(n) = (p-1)(q-1) = 3120. 

What bothers me is that when I try to generate the public exponent e using CrypTool (given the same parameters), it gives me e = 2^16+1 = 65537 BUT I thought e had to be less that phi(n) = (p-1)(q-1) (according to this source).
Why is it OK to break this rule? Or is the CrypTool faulty?
The $2^{16} + 1$ exponent is really meant for use in real life systems, in which public keys are expected to be considerably larger than that. I guess CrypTool assumes this is the case, as you would expect, really.

That said, if $\gcd{(e, \varphi{(n)})} = 1$, then $gcd{(e ~ \mathrm{mod} ~ \varphi{(n)}, \varphi{(n)})} = 1$ by definition, so such an $e$ will still work regardless, as noted by fgrieu in the comments.