## Digital root sums of factorisations.

### Problem 159

Published on Saturday, 30th June 2007, 02:00 pm; Solved by 1635

A composite number can be factored many different ways. For instance, not including multiplication by one, 24 can be factored in 7 distinct ways:

24 = 2x2x2x3
24 = 2x3x4
24 = 2x2x6
24 = 4x6
24 = 3x8
24 = 2x12
24 = 24

Recall that the digital root of a number, in base 10, is found by adding together the digits of that number, and repeating that process until a number is arrived at that is less than 10. Thus the digital root of 467 is 8.

We shall call a Digital Root Sum (DRS) the sum of the digital roots of the individual factors of our number.
The chart below demonstrates all of the DRS values for 24.

Factorisation | Digital Root Sum
--- | ---
2x2x2x3 | 9
2x3x4 | 9
2x2x6 | 10
4x6 | 10
3x8 | 11
2x12 | 5
24 | 6

The maximum Digital Root Sum of 24 is 11.
The function mdrs(n) gives the maximum Digital Root Sum of n. So mdrs(24)=11.
Find mdrs(n) for 1 n 1,000,000.