Show that for each positive integer n equal to twice a triangular number, the corresponding expression represents an integer.
Let n be such an integer, then there exists a positive integer m such that . We then have so that we have successively
; ; and so on. It follows that
, as required.
Comment: you have to be a bit aware of properties of triangular numbers.
1001 Problems in Classical Number Theory by Jean-Marie De Koninck and Armel Mercier, AMS (American Mathematical Society), Indian Edition:
Amazon India link: