Multiplicative arithmetic is the first-order theory T of natural 
numbers with multiplication. A famous theorem of Skolem (1930) asserts that 
T is decidable. Later proofs of this fact were given by Feferman-Vaught
(1959) and Ferrante-Rackoff (1979). The latter also established an asymptotic
upper complexity bound for the decision problem. All proofs are based on the 
isomorphism of $(N,.)$ with $(N_0^*,+)$ and the decidability of 
Presburger arithmetic. 

The talk investigates the problem of algorithmic quantifier elimination
(QE) for T in a suitable extension language. In his Habilitationsschrift (1979)
the author established a QE procedure for T in a large and rather 
unnatural extension language by very general principles. Here we present 
an efficient  "restricted" QE for T can be performed in a "natural" extension
language. The method yields also explicit parametric solutions for 
existential problems. We also discuss the asymptotic and the practical 
complexity of the method.