MathDB

IMO Shortlist 2011, Number Theory 1

Source: IMO Shortlist 2011, Number Theory 1

July 11, 2012

algorithmnumber theoryprime numbersDivisibilityIMO Shortlistnumber of divisors

Problem Statement

For any integer

d > 0,

let

f(d)

be the smallest possible integer that has exactly

d

positive divisors (so for example we have

f(1)=1, f(5)=16,

and

f(6)=12

). Prove that for every integer

k \geq 0

the number

f\left(2^k\right)

divides

f\left(2^{k+1}\right).

Proposed by Suhaimi Ramly, Malaysia