MathDB

Start with a finite sequence

a_1,a_2,\dots,a_n

of positive integers. If possible, choose two indices

j < k

such that

a_j

does not divide

a_k

and replace

a_j

and

a_k

\gcd(a_j,a_k)

and

\text{lcm}\,(a_j,a_k),

respectively. Prove that if this process is repeated, it must eventually stop and the final sequence does not depend on the choices made. (Note:

\gcd

means greatest common divisor and lcm means least common multiple.)

A3