MathDB

A carpet dealer,who has a lot of carpets in the market,is available to exchange a carpet of dimensions

a\cdot b

either with a carpet with dimensions

\frac{1}{a}\cdot \frac{1}{b}

or with two carpets with dimensions

c\cdot b

and

\frac{a}{c}\cdot b

(the customer can select the number

c

).The dealer supports that,at the beginning he had a carpet with dimensions greater than

1

and,after some exchanges like the ones we described above,he ended up with a set of carpets,each one having one dimension greater than

1

and one smaller than

1

.Is this possible?
Note:The customer can demand from the dealer to consider a carpet of dimensions

a\cdot b

as one with dimensions

b\cdot a

1