#274 - Divisibility Multipliers
For each integer \(p \gt 1\) coprime to \(10\) there is a positive divisibility multiplier \(m \lt p\) which preserves divisibility by \(p\) for the following function on any positive integer, \(n\):
\(f(n) = (\text{all but the last digit of }n) + (\text{the last digit of }n) \cdot m\).
That is, if \(m\) is the divisibility multiplier for \(p\), then \(f(n)\) is divisible by \(p\) if and only if \(n\) is divisible by \(p\).
(When \(n\) is much larger than \(p\), \(f(n)\) will be less than \(n\) and repeated application of \(f\) provides a multiplicative divisibility test for \(p\).)
For example, the divisibility multiplier for \(113\) is \(34\).
\(f(76275) = 7627 + 5 \cdot 34 = 7797\): \(76275\) and \(7797\) are both divisible by \(113\).
\(f(12345) = 1234 + 5 \cdot 34 = 1404\): \(12345\) and \(1404\) are both not divisible by \(113\).
The sum of the divisibility multipliers for the primes that are coprime to \(10\) and less than \(1000\) is \(39517\). What is the sum of the divisibility multipliers for the primes that are coprime to \(10\) and less than \(10^7\)?
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=274. Published Friday, 15th January 2010, 01:00 pm. Solved by 1,694 members at time of mirroring.
Why this is useful
Mathematical Foundation. Exact integer reasoning and algorithmic efficiency. Foundational rigour and computational discipline rather than a direct trading application.
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Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.7 Dynamic Programming: Memoization and Tabulation · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.2 Primes, Sieves, and Integer Factorization
Recommended stepping-stone problems: #78 · #179 · #187
Concepts: number-theory
Likely techniques: digit-dp gcd-euclid sieve
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- What exactly is the input to problem 274? Is it a bound (10^7), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single sum.
- Which objects exactly are in scope, and which are excluded by the wording (strict vs non-strict inequality, 'distinct', 'proper', 'below' vs 'up to')?
- What do the arguments of f(n) mean, and what is the value's type (count, sum, probability)?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 10^7?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 10^7 and the cost of testing one.
- Which number-theory fact would, if true, collapse the search - and can you state it as a testable claim before you look for a proof?
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- Predict the strategy: in one sentence, what will your solution do? (The classification says number-theory / digit-dp - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 10^7, and the wall-clock time you expect. Write both down now.
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- Predict the failure mode: what is most likely to break - an off-by-one on the bound, a definition misread, precision, or memory?
- Predict the output of the small case from rung 3 BEFORE running it (the statement says: "For example, the divisibility multiplier for 113 is 34.") - then run it. A surprise here is worth more than an hour of debugging later.
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- Related problem: #78 · #179 · #187
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