Problem:

Show that for each positive integer n equal to twice a triangular number, the corresponding expression represents an integer.

Solution:

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.

Reference:

1001 Problems in Classical Number Theory by Jean-Marie De Koninck and Armel Mercier, AMS (American Mathematical Society), Indian Edition:

Amazon India link:

https://www.amazon.in/1001-Problems-Classical-Number-Theory/dp/0821868888/ref=sr_1_1?s=books&ie=UTF8&qid=1508634309&sr=1-1&keywords=1001+problems+in+classical+number+theory

Cheers,

Nalin Pithwa.