Start with a finite sequence a1, a2, . . . , an of positive

integers. If possible, choose two indices j < k such

that aj does not divide ak, and replace aj and ak by

gcd(aj , ak) and lcm(aj , ak), 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.)