Problem D

Digits Multiplied

The digits of any positive integer can be multiplied together to give
the *digit product* for the integer. For example, \(345\) has the digit product of \(3 \times 4 \times 5 = 60\). There are many
other positive integers that have \(60\) as a digit product. For example, \(2532\) and \(14\,153\) both have a digit product of
\(60\). Note that \(256\) is the smallest positive integer with
a digit product of \(60\).

There are also many positive integers that have a digit product of \(2160\). Determine the smallest such integer.