#159 - Digital Root Sums of Factorisations
A composite number can be factored many different ways. For instance, not including multiplication by one, \(24\) can be factored in \(7\) distinct ways:
\[\begin{align} 24 &= 2 \times 2 \times 2 \times 3\\ 24 &= 2 \times 3 \times 4\\ 24 &= 2 \times 2 \times 6\\ 24 &= 4 \times 6\\ 24 &= 3 \times 8\\ 24 &= 2 \times 12\\ 24 &= 24 \end{align}\]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 |
|---|---|
| \(2 \times 2 \times 2 \times 3\) | \(9\) |
| \(2 \times 3 \times 4\) | \(9\) |
| \(2 \times 2 \times 6\) | \(10\) |
| \(4 \times 6\) | \(10\) |
| \(3 \times 8\) | \(11\) |
| \(2 \times 12\) | \(5\) |
| \(24\) | \(6\) |
The maximum Digital Root Sum of \(24\) is \(11\).
The function \(\operatorname{mdrs}(n)\) gives the maximum Digital Root Sum of \(n\). So \(\operatorname{mdrs}(24)=11\).
Find \(\sum \operatorname{mdrs}(n)\) for \(1 \lt n \lt 1\,000\,000\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=159. Published Saturday, 30th June 2007, 02:00 pm. Solved by 3,888 members at time of mirroring.
Why this is useful
Mathematical Foundation. Exact counting underlies discrete pricing lattices, scenario enumeration, and combinatorial probability (Phase 7).
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Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 1.1 Sets, Functions, and Relations · 17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.7 Dynamic Programming: Memoization and Tabulation · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.11 Integer Partitions and Counting Structures · 19.2 Primes, Sieves, and Integer Factorization · 2.1 Functions, Limits, and Continuity · 4.2 Linear Maps, Matrices, Rank, and the Null Space · 7.1 Probability Spaces, Random Variables, and Distributions
Recommended stepping-stone problems: #309 · #66 · #243
Concepts: algebra combinatorics number-theory brute-force-reduction
Likely techniques: digit-dp factorization hashing
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- Related problem: #309 · #66 · #243
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