Problem
ALG-B1-M05-P021 Almost Rectangular Numbers
#21
★★★★★ Level 5 of 5
Call a number rectangular if it equals \(m(m+1)\). Prove that every rectangular number can be represented as a quotient of two rectangular numbers.
Inspired by final olympiad method · 2015 · Grade 10 · Problem 1
Try multiplying \(n(n+1)\) by \(\frac{(n+1)(n+2)}{(n+1)(n+2)}\).
Let the given number be \(n(n+1)\). Then
\[n(n+1)=\frac{n(n+2)(n+1)^2}{(n+1)(n+2)}=\frac{(n^2+2n)(n^2+2n+1)}{(n+1)(n+2)}.\]
The numerator is the product of consecutive integers \(n^2+2n\) and \(n^2+2n+1\), and the denominator \((n+1)(n+2)\) is also rectangular. Thus the representation is found.
This is a sequence-style construction: from \(n\), we build a new pair of consecutive numbers.