Find the smallest five-digit natural number whose digits product is $3\, 024$.
Simone solved the task in the following steps:
(1) She found a prime factor decomposition of the number $3\, 024$: $$\begin{aligned} 3 024&=2\cdot1\, 512=\cr &=2\cdot2\cdot756= \cr &=2\cdot2\cdot2\cdot378=\cr &=2\cdot2\cdot2\cdot2\cdot189=\cr &=2\cdot2\cdot2\cdot2\cdot3\cdot63=\cr &=2\cdot2\cdot2\cdot2\cdot3\cdot3\cdot21=\cr &=2\cdot2\cdot2\cdot2\cdot3\cdot3\cdot3\cdot7\cr \end{aligned}$$
(2) She rewrote the result as the product of five one-digit factors: $$3\, 024=2^4\cdot3^3\cdot7^1=2\cdot2^3\cdot3\cdot3^2\cdot7^1$$
(3) She claimed that the five smallest digits whose product is $3\, 024$ are: $$2, 8, 3, 9, 7$$
(4) Simone ordered the digits in ascending order and claimed that the smallest five-digit natural number whose digit product is $3\, 024$ is: $$23\, 789$$ Is Simone's solution correct? If not, identify where Simone made a mistake in the procedure.
Simone's solution is correct.
The mistake is in step (1). Simone made a mistake in prime factor decomposition of the number $3\, 024$.
The mistake is in step (3). Simone did not choose correct digits for the sought 5-digit number to make it as small as possible.
The mistake is in step (4). Simone did not write the smallest possible natural number from the found digits.
Simone did not realize that the number $1$ is not included in the prime number decomposition. While it is not a prime number, it is both a digit and a number. Therefore: $$3\, 024=1\cdot2^4\cdot3^3\cdot7^1=1\cdot(2\cdot3)\cdot 2^3\cdot3^2\cdot7^1$$ The five smallest digits whose product is $3\, 024$ are: $$1, 6, 8, 9, 7.$$ When arranged in ascending order, we get the smallest five-digit natural number, $16\, 789$, whose digit product is $3\, 024$.