Smallest number with digit product 10000

This brain teaser is about constructing an integer from its digit-level structure, under a multiplicative constraint. You must connect number theory, base-10 representation, and combinatorics to understand what sets of digits are even feasible. It sits right at the intersection of recreational math and the kind of discrete reasoning that shows up in real quant interviews.

It trains prime factorization, comfort with digits as "building blocks," and numerical optimization under constraints. You practice turning a high-level requirement into a search over structured possibilities, pruning bad candidates quickly, and reasoning about minimality without brute-force enumeration. This is exactly the style of mental agility good quant prep should emphasize.

It matters for quant interviews because it exposes whether you can systematize a puzzle, not just tinker. Interviewers see how you manage constraints, argue optimality, and keep arithmetic clean under time pressure.

What it tests

Whenever you are asked to find a number whose digits multiply to a given product, the key is to factor the target product into primes and then group those primes into allowable digits (2 through 9) in a way that minimizes the resulting number. This is because each digit can only be a single number from 1 to 9, so you must represent the product using only these. The guiding principle is to use the largest possible digits that represent multiple small primes (like 8 for $2^3$ or 9 for $3^2$), since fewer, larger digits generally produce smaller numbers when arranged in ascending order. The reason this works is that minimizing the number of digits and keeping the largest digits to the right (least significant place) reduces the overall value of the integer.