a) 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 1996

Number theory

Easy

Challenge and Thrill of Pre-college Mathematics

Do you really need a hint? Try it first!

Note that, if if and only if .

For in a finite set , we can simply choose (observing hint 1) . This gives .

If is infinite, the integer is required to be divisible by for arbitrarily large . That is, is required to have infinitely many (and arbitrarily large) divisors. This cannot happen unless . Hence in this case.

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