Solution
The Stubborn Product
Show the problem again
What is the smallest positive integer whose decimal digits have product equal to 10! = 3 628 800? Give the integer.
Worked solution
Since 10! = 2⁸ · 3⁴ · 5² · 7, any digit 0 or 1 is useless, and the digits must absorb the primes 5, 5 and 7 as the digits 5, 5, 7. For the remaining 2⁸ · 3⁴, fewer digits mean a smaller number, so pack greedily: 9 · 9 takes the 3⁴, and 8 · 8 · 4 takes the 2⁸. That gives eight digits, and no seven-digit arrangement exists because the largest available digits cannot hold 2⁸ in fewer than three digits. Sorting the multiset {4, 5, 5, 7, 8, 8, 9, 9} ascending gives the smallest number, **45578899**. An exhaustive search over all digit multisets agrees.
Source: Digit-product minimisation, a standard recreational and olympiad-training theme; this instance written for AxiomIQ.