Find a three digit number that fits the following criteria:

1)The digits are all different.

2)The product of the digits times the largest of the 3 digits equals the original 3 digit number.

3)The digits are either all odd or all even.

An n-digit number is assumed to be zero-leading and base-10, yet if not limited to base-10 there are other possible solutions.

The smallest base where there is a solution is base-7: 135, which like the base-10 solution has all odd digits.
The smallest base where there is a solution for all even digits is base-31: C2M. 
The smallest base where there are two or more solutions is base 35 and having two solutions:  2CA and M4E, both with digits of even parity.
The smallest base where there are multiple solutions of differing parity is base 43: 4AE (even) PF5 (odd).
The smallest base where there are three or more solutions is base 77 having three solutions, each having odd parity: 1aB, 1q5, and 3e7.

Note: A base85 system of digits has been proposed as of April 1, 1996 with the standard set of digits 0~9, A~Z, followed by the extended a~z, and extended by the 23-digits of ASCII characters:  ! # $ % & ( ) * + - ; < = > ? @ ^ _ ` { | } ~  
 
The smallest base where one of the 23 special characters is part of the solution is base 93: 42&  [even parity].
The proposed base85 system is insufficient as a single character per digit display of all of base 93's digits, but there is no other solution for this base -- or any base less than 100 -- requiring any further extension.  

Edited on September 17, 2020, 9:51 am

Posted by Dej Mar on 2020-09-17 09:11:31