729. Does there exist a natural number $n$ such that the number $3 n+1$ can be represented as the product of two natural numbers?
$\triangle$ The number $3n+1$ when divided by 3 gives a remainder of 1, while the number $k(k+1)$ for a natural number $k$ gives a remainder of 0 or 2 (prove it!). Therefore, such a natural number $n$ does not exist.

Answer: does not exist.