TITLE: Any natural number is greater than or equal to its product of digits.
QUESTION [30 upvotes]: I was randomly thinking and I stumbled upon this question in my mind. I was thinking of numbers, and I observed that any number is greater than or equal to the product of digits in many cases. I was thinking, is this true for all natural numbers or for only a few cases? Please help.

REPLY [6 votes]: $\begin{eqnarray} {\bf Hint}
& & a + 10\ b + 10^2 c + 10^3 d\\
&=&   a+\color{#c00}{10}(b+\color{#c00}{10}(c+\color{#c00}{10}d))\\
&\ \ge\ & a+\ \color{#c00}a\ (b +\ \color{#c00}b\ (c+\ \color{#c00}c\ d))\quad\ \ {\rm by}\ \ \ \color{#c00}{10 > a,b,c}\\
& =& a+\ a\,\  b + a\ b\ c+ abcd\  \ge\ abcd
\end{eqnarray}$