Let $T$ be the set of all positive factor of $n= 2004^{2010}$. Suppose that $S$ be an arbitrary nonempty subset of $T$ satisfying the fact of all $a$, $b$ belong to $S$ and $a>b$ then $a$ not divisible $b$. Find the maximal number of elements of such subset $S$
I don't have any idea for this problem. It's very interesting but strange.
Hint: Members of $T$ are of the form $2^r 3^s 167^t$ where $0 \le r \le 4020$, $0 \le s \le 2010$ and $0 \le t \le 2010$. If $a = 2^r 3^s 167^t$ and $b = 2^u 3^v 167^w$ are distinct members of $S$, you need at least one of $r-u$, $s-v$, $t-w$ to be strictly positive and at least one strictly negative. You might consider $S$ of the form $\{2^r 3^s 167^t\; : 0 \le r\le 4020,\; 0 \le s\le 2010,\; 0 \le t\le 2010,\; r+s+t = m\}$ for suitable $m$.