MathDB

4

Part of 2015 Brazil National Olympiad

Problems(1)

Divisors

Source: Problem 4, Brazilian MO 2015

10/20/2015
Let nn be a integer and let n=d1>d2>⋯>dk=1n=d_1>d_2>\cdots>d_k=1 its positive divisors. a) Prove that d1−d2+d3−⋯+(−1)k−1dk=n−1d_1-d_2+d_3-\cdots+(-1)^{k-1}d_k=n-1 iff nn is prime or n=4n=4. b) Determine the three positive integers such that d1−d2+d3−...+(−1)k−1dk=n−4.d_1-d_2+d_3-...+(-1)^{k-1}d_k=n-4.
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