735 is a 3-digit base ten positive integer formed by a permutation of the digits 3, 5 and 7 and, 735 is divisible by each of 3, 5 and 7.

Determine the minimum value of N with N ≥ 11 such that there exists a 3-digit base-N positive integer p which is formed by a permutation of the digits 3, 5 and 7 such that p is divisible by each of 3, 5 and 7. What is the next smallest value of N with this property?

The following table is a list of the permutations of 3-digit positive integers formed from the permutations of the numerals 3,5 and 7, with the smallest integer base of unmixed radix (closest to zero) followed by the smallest integer base of unmixed radix greater than 10 for the given number where the number is divisible by 3, 5 and 7,

357: -11; 31
375: (no unmixed integer radix found)
537: -14; 26
573: (no unmixed integer radix found)
735: 10; 31
753: -9; 19

The smallest (closest to zero) is the neganonary base (-9) for 753 (525 base 10), yet the minimum value sought is for the unmixed integer radix > 10, thus the answer is 19 for 753 (2625 base 10), and the next smallest is 26 for 537 (3465 base 10).

Posted by Dej Mar on 2013-03-04 09:10:42