MathDB

Let

T

be a finite set of positive integers, satisfying the following conditions: 1. For any two elements of

T

, their greatest common divisor and their least common multiple are also elements of

T

. 2. For any element

x

T

, there exists an element

x'

T

such that

x

and

x'

are relatively prime, and their least common multiple is the largest number in

T

.
For each such set

T

, denote by

s(T)

its number of elements. It is known that

s(T) < 1990

; find the largest value

s(T)

may take.

Problem Statement