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INMO 1996 Problem 1

Understand the problem

a) Given any positive integer $n$, show that there exist distint positive integers $x$ and $y$ such that $x + j$ divides $y + j$ for $j = 1 , 2, 3, \ldots, n$;

b) If for some positive integers $x$ and $y$, $x+j$ divides $y+j$ for all positive integers $j$, prove that $x = y$.

Source of the problem
Indian National Mathematical Olympiad 1996
Topic
Number theory
Difficulty Level
Easy
Suggested Book
Challenge and Thrill of Pre-college Mathematics

Start with hints

Do you really need a hint? Try it first!

Note that, if x+j | y+j if and only if x+j | y-x.

For j in a finite set I, we can simply choose (observing hint 1) y-x=\prod_{j\in I}(x+j). This gives y=x+\prod_{j\in I}(x+j).

If I is infinite, the integer y-x is required to be divisible by x+j for arbitrarily large j. That is, y-x is required to have infinitely many (and arbitrarily large) divisors. This cannot happen unless y-x=0. Hence x=y in this case.

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