<< problem 128 - Hexagonal tile differences | Composites with prime repunit property - problem 130 >> |

# Problem 129: Repunit divisibility

(see projecteuler.net/problem=129)

A number consisting entirely of ones is called a repunit. We shall define R(k) to be a repunit of length k; for example, R(6) = 111111.

Given that n is a positive integer and GCD(n, 10) = 1, it can be shown that there always exists a value, k, for which R(k) is divisible by n,

and let A(n) be the least such value of k; for example, A(7) = 6 and A(41) = 5.

The least value of n for which A(n) first exceeds ten is 17.

Find the least value of n for which A(n) first exceeds one-million.

# My Algorithm

Only if x is a divisor of y then y mod x == 0.

My code generates all repunits R(k) until R(k) mod x == 0:

R(k+1) = R(k) * 10 + 1

Unfortunately, R(20) is already too big for a 64 bit integer.

But I don't actually need to work with R(k), I can use its mod x, too:

R(k) mod x = R_{mod x}(k) mod x

R_{mod x}(k+1) = (R_{mod x}(k) * 10 + 1) mod x

x fits easily in a 32 bit integer, therefore the "modulo repunit" fits in there, too.

Initially I started a brute-force search with `i = 1`

which took a few minutes. However, it seems that always A(i) <= i.

## Modifications by HackerRank

Apparently there is a much smarter way to solve this - that I have no idea about. All but the first test case time out.

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

The code contains `#ifdef`

s to switch between the original problem and the Hackerrank version.

Enable `#ifdef ORIGINAL`

to produce the result for the original problem (default setting for most problems).

#define ORIGINAL
#include <iostream>
// return minimum k where R(k) is divisible by x

unsigned long long getMinK(unsigned long long x)
{
// same as gcd(x, 10) = 1
if (x % 5 == 0 || x % 2 == 0)
return 0;
// "number of ones"
unsigned long long result = 1;
// current repunit mod divisor
unsigned long long repunit = 1;
// no remainder ? that repunit can be divided by divisor
while (repunit != 0)
{
// next repunit
repunit *= 10;
repunit++;
// keep it mod divisor
repunit %= x;
result++;
}
return result;
}
int main()
{
unsigned int tests;
std::cin >> tests;
while (tests--)
{
// I observed that getMinK(i) <= i
// I lack the mathematic skills to prove (or disprove) it
// at least it holds up to the result when running a brute-force search beginning at 1
unsigned long long limit = 1000000;
std::cin >> limit;
unsigned long long i = limit;
#ifdef ORIGINAL
while (getMinK(i) <= limit)
i++;
std::cout << i << std::endl;
#else
std::cout << getMinK(i) << std::endl;
#endif
}
return 0;
}

This solution contains 7 empty lines, 10 comments and 5 preprocessor commands.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

This is equivalent to`echo "1 10" | ./129`

Output:

*Note:* the original problem's input `1000000`

__cannot__ be entered

because just copying results is a soft skill reserved for idiots.

*(this interactive test is still under development, computations will be aborted after one second)*

# Benchmark

The correct solution to the original Project Euler problem was found in 0.02 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

(compiled for x86_64 / Linux, GCC flags: `-O3 -march=native -fno-exceptions -fno-rtti -std=c++11 -DORIGINAL`

)

See here for a comparison of all solutions.

**Note:** interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without `-DORIGINAL`

.

# Changelog

May 21, 2017 submitted solution

May 22, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler129

My code solves **1** out of **7** test cases (score: **0%**)

I failed **0** test cases due to wrong answers and **6** because of timeouts

# Difficulty

Project Euler ranks this problem at **45%** (out of 100%).

Hackerrank describes this problem as **hard**.

*Note:*

Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.

In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is rarely an option.

# Similar problems at Project Euler

Problem 130: Composites with prime repunit property

*Note:* I'm not even close to solving all problems at Project Euler. Chances are that similar problems do exist and I just haven't looked at them.

# Links

projecteuler.net/thread=129 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/project-euler-129-minimal-repunits/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p129.java (written by Nayuki)

# Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.

yellow problems score less than 100% at Hackerrank (but still solve the original problem).

gray problems are already solved but I haven't published my solution yet.

blue problems are solved and there wasn't a Hackerrank version of it at the time I solved it or I didn't care about it because it differed too much.

red problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte.

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I scored 13,183 points (out of 15300 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

Look at my progress and performance pages to get more details.

My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

# Copyright

I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.

All of my solutions can be used for any purpose and I am in no way liable for any damages caused.

You can even remove my name and claim it's yours. But then you shall burn in hell.

The problems and most of the problems' images were created by Project Euler.

Thanks for all their endless effort.

<< problem 128 - Hexagonal tile differences | Composites with prime repunit property - problem 130 >> |