## Digital root sums of factorisations

### Problem 159

Published on Saturday, 30th June 2007, 02:00 pm; Solved by 1976; Difficulty rating: 60%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

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.