# How many possible values does the product of $n$ numbers take?

Let $S = \{1,2,...n\}$ and $s$, an element of $S$. Similarily, $T = \{1,2,...m\}$ and $t$ an element of $T$. My question is, how many values of $st$ are there? More generally, if I am given $x$ such sets how many values of $stuv...$ are there (multiplying over the $x$ elements).

Motivation: I was trying to solve a more general version of the $1990$ Putnam $B3$ which involved squaring numbers of form $st$ after multiplying matrices.

• The set-product of the two sets would be a set of ordered pairs, of cardinality $mn$, correct? If you map an ordered pair to the product of its ordinate and abscissa, it would be a one - to - one mapping. So the number of products would match, and you would have $mn$ products, though some would undoubtedly multiply to the same number. Are you asking how many different numbers the product could be? – AmateurMathPirate Aug 28 '17 at 0:47
• Yes, otherwise it would easily be $mn$. – mtheorylord Aug 28 '17 at 1:23
• Just wanted to be sure. The mystery is tied to the mystery of the primes i think. – AmateurMathPirate Aug 28 '17 at 1:25

I don't know, but for $m=n$ and $n<1000$, it seemed to fit $$0.65\frac{n^2}{\sqrt{\ln n}}$$