29 Aug 2013

IrMO 1997 P2 Q1: Abundant Numbers: Upper Secondary Mathematics Competition Question

Given a positive integer n, denote by sigma(n) the sum of all positive integers which divide n. [For
example, sigma(3) = 1 + 3 = 4, sigma(6) = 1 + 2 + 3 + 6 = 12, sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28].

We say that n is abundant if sigma(n) > 2n. (So, for example, 12 is abundant).

Let a, b be positive integers and suppose that a is abundant. Prove that ab is abundant.

[IrMO 1997 Paper 2 Question 1]

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