  A pair of natural numbers (a,b) such that a is both squarefree and coprime to
b is called a carefree couple.
  A result conjectured by Manfred Schroeder (in his book `Number theory in
science and communication') on carefree couples and a variant of it are
established using standard arguments from elementary analytic number theory.
Also a related conjecture of Schroeder on triples of integers that are pairwise
coprime is proved.
