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Two ugly numbers don't make them beautiful

Source: Indonesian Mathematical Olympiad 2014 Day 2 Problem 8

September 4, 2014
modular arithmeticnumber theory unsolvednumber theory

Problem Statement

A positive integer is called beautiful if it can be represented in the form x2+y2x+y\dfrac{x^2+y^2}{x+y} for two distinct positive integers x,yx,y. A positive integer that is not beautiful is ugly.
a) Prove that 20142014 is a product of a beautiful number and an ugly number. b) Prove that the product of two ugly numbers is also ugly.
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