1) Find all the 0 to 9 pandigital numbers (without leading zero) that have the largest power of 3 as a factor.

2) One of these numbers has a very interesting property. What is it?

*an x to y pandigital number is an integer that contains all the digits from x to y and only those digits once each, for example 1234 is 1 to 4 pandigital but not 1 to 9 pandigital*

The pandigital number with the largest power of 3 as a factor is 7246198035 with the factors 3¹⁵ x 5 x 101.

The eight other pandigital numbers with the factor 3¹⁴ are:
3410256897 = 3¹⁴ x 23 x 31
5361708249 = 3¹⁴ x 19 x 59
5902183746 = 3¹⁴ x 2 x 617
6820513794 = 3¹⁴ x 2 x 23 x 31
8145396207 = 3¹⁴ x 13 x 131
8269753401 = 3¹⁴ x 7 x 13 x 19
9145036728 = 3¹⁴ x 2³ x 239
9537240186 = 3¹⁴ x 2 x 997

In regard to "interesting properties", one of the factors of the pandigital number 5902183746 is that it has a factor that is a number with digits of consecutive sequence, i.e., 1234 [2 x 617]; another is the pandigital number 8269753401 with the palindromic factors 19 and 91 [7 x 13]; but most likely, the most interesting may be that the pandigital number 6820513794 has the pandigital number 3410256897 [3¹⁴ x 23 x 31] as a factor.

 

Edited on February 24, 2009, 7:09 pm

Posted by Dej Mar on 2009-02-24 17:04:24