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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 $x = y$.

##### Source of the problem
Indian National Mathematical Olympiad 1996
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.

# Connected Program at Cheenta

#### Math Olympiad Program

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year.

Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

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