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4 digits number using only 2 disticnt digits problem

Source: Austrian Polish 1984 APMC

April 30, 2020

GCDgreatest common divisornumber theory

Problem Statement

Let

A

be the set of four-digit natural numbers having exactly two distinct digits, none of which is zero. Interchanging the two digits of

n\in A

yields a number

f (n) \in A

(for instance,

f (3111) = 1333

). Find those

n \in A

with

n > f (n)

for which

gcd(n, f (n))

is the largest possible.