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The number of positive divisors

Source: Middle European Mathematical Olympiad 2012 - Team Compt. T-8

September 14, 2012

number theory proposednumber theory

Problem Statement

For any positive integer

n

let

d(n)

denote the number of positive divisors of

n

. Do there exist positive integers

a

and

b

, such that

d(a)=d(b)

and

d(a^2 ) = d(b^2 )

, but

d(a^3 ) \ne d(b^3 )