A month ago, I took a test and was asked whether “a group of 689
elements is closed under the group operation”. The question is
*trivial*, but one can discover another question: **Does such a group
exists?**

In order to “proceed with with my work”, I *temporarily accepted* the
existence fo swuch a group during the test. However, even though I
got the question right and earned some marks for that, I *didn’t*
think that this is *true* mathematics. To say that one *knows* this
quoted statement, one has to follow the *“international standard”* of
a group and make *rigourous arguements* from the *“basic laws”* to
prove the existence of such a group.

Thinking that the group of 689 elements *isn’t* useful, I put that
problem aside. After a fortnight, I came up with an answer:
$\Z_{13} \times \Z_{53} \cong
\Z_{689}$.