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# 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 $x = y$.

Number theory
Easy
##### Suggested Book
Challenge and Thrill of Pre-college Mathematics

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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# Similar Problems

## INMO 1996 Problem 1

Understand the problema) Given any positive integer , show that there exist distint positive integers and such that divides for ; b) If for some positive integers and , divides for all positive integers , prove that .Indian National Mathematical Olympiad...