MathDB

Israel 2015 Q4 - Divisibility conditions

Source: Israel National Olympiad 2015 Q4

August 7, 2019

number theoryDivisibility

Problem Statement

Let

k,m,n

be positive integers such that

n^m

is divisible by

m^n

, and

m^k

is divisible by

k^m

.
[*] Prove that

n^k

is divisible by

k^n

. [*] Find an example of

k,m,n

satisfying the above conditions, where all three numbers are distinct and bigger than 1.