1998 IMO Problems/Problem 4
Determine all pairs of positive integers such that divides .
We use the division algorithm to obtain Here is a solution of the original statement, possible when and where is any natural number. This is easily verified.
Otherwise we obtain the inequality (by basic properties of divisiblity): So
Testing for we find that Therefore, , and we can easily check these.
Testing for and applying the division algorithm we find that , having no solutions in natural .
Hence, the only solutions are: for all natural .
Written by dabab_kebab
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