IBM Ponder This #318 · October 2024

This puzzle was suggested by Karl Heinz Hofmann - thanks!

We are looking for a natural number X satisfying the following properties of its decimal representation:

1. All the digits $\{1,2,3,4,6,7,8,9\}$ appear at least once, but $\{0,5\}$ do not appear at all.

2. X can be written as a front part A and back part B, such that if X were a string, it could be written as A concatenated with B, such that

3. B is the product of the digits of X.

4. X is a multiple of B.

5. B is a perfect square.

For example, $X=3411296$ has the property that its digit product is $1296$ so it can be split into $A=341$ and $B=1296$ such that properties 2.1 and 2.3 are satisfied. However, none of the other properties are satisfied in this case since $X$ is missing the digits $7,8$ and is not a multiple of $B$.

Your goal: Find the least number X satisfying the above properties.

A bonus "*" will be given for finding the least number X satisfying the above properties except property 2.3 which is replaced by a new property: 2.3 The digit product of A is a perfect cube.

Solution

Best opened after a real attempt

To be added.