<< problem 132 - Large repunit factors | Prime pair connection - problem 134 >> |

# Problem 133: Repunit nonfactors

(see projecteuler.net/problem=133)

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.

Let us consider repunits of the form R(10^n).

Although R(10), R(100), or R(1000) are not divisible by 17, R(10000) is divisible by 17.

Yet there is no value of n for which R(10^n) will divide by 19. In fact, it is remarkable that 11, 17, 41, and 73 are the only four primes below one-hundred that can be a factor of R(10^n).

Find the sum of all the primes below one-hundred thousand that will never be a factor of R(10n).

# My Algorithm

After solving problem 132 I played around with the code and just got lucky:

I tried looping over several repunits but it turned out it is sufficient to test only one really, really large repunit.

Unfortunately, I cannot give any satisfying reasoning. I saw some proves on other websites, though.

In order to work with such "really, really large repunit" I had to swap my `powmod`

code from problem 132 with code that can handle 64 bit values.

It's part of my toolbox, too.

# My code

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

#include <iostream>
#include <vector>
// return (a*b) % modulo

unsigned long long mulmod(unsigned long long a, unsigned long long b, unsigned long long modulo)
{
// (a * b) % modulo = (a % modulo) * (b % modulo) % modulo
a %= modulo;
b %= modulo;
// fast path
if (a <= 0xFFFFFFF && b <= 0xFFFFFFF)
return (a * b) % modulo;
// we might encounter overflows (slow path)
// the number of loops depends on b, therefore try to minimize b
if (b > a)
std::swap(a, b);
// bitwise multiplication
unsigned long long result = 0;
while (a > 0 && b > 0)
{
// b is odd ? a*b = a + a*(b-1)
if (b & 1)
{
result += a;
result %= modulo;
// skip b-- because the bit-shift at the end will remove the lowest bit anyway
}
// b is even ? a*b = (2*a)*(b/2)
a <<= 1;
a %= modulo;
// next bit
b >>= 1;
}
return result;
}
// return (base^exponent) % modulo

unsigned long long powmod(unsigned long long base, unsigned long long exponent, unsigned long long modulo)
{
unsigned long long result = 1;
while (exponent > 0)
{
// fast exponentation:
// odd exponent ? a^b = a*a^(b-1)
if (exponent & 1)
result = mulmod(result, base, modulo);
// even exponent ? a^b = (a*a)^(b/2)
base = mulmod(base, base, modulo);
exponent >>= 1;
}
return result;
}
int main()
{
// a large exponent for powmod => 10^(10^19)
const unsigned long long digits = 10000000000000000000ULL;
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
unsigned int maxPrime = 100000;
std::cin >> maxPrime;
unsigned long long sum = 0;
std::vector<unsigned int> primes;
for (unsigned int i = 2; i < maxPrime; i++)
{
bool isPrime = true;
// test against all prime numbers we have so far (in ascending order)
for (auto x : primes)
{
// prime is too large to be a divisor
if (x*x > i)
break;
// divisible => not prime
if (i % x == 0)
{
isPrime = false;
break;
}
}
// no prime
if (!isPrime)
continue;
primes.push_back(i);
// check for divisibility by 9*prime
auto modulo = 9 * i;
// remainder must not be 1
auto remainder = powmod(10, digits, modulo);
if (remainder != 1)
sum += i;
}
std::cout << sum << std::endl;
}
return 0;
}

This solution contains 19 empty lines, 21 comments and 2 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 "1 100" | ./133`

Output:

*Note:* the original problem's input `100000`

__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

June 2, 2017 submitted solution

June 2, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler133

My code solves **9** out of **11** test cases (score: **80%**)

I failed **0** test cases due to wrong answers and **2** because of timeouts

# Difficulty

Project Euler ranks this problem at **50%** (out of 100%).

Hackerrank describes this problem as **medium**.

*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.

# Links

projecteuler.net/thread=133 - **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-133-repunit-nonfactors/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p133.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 132 - Large repunit factors | Prime pair connection - problem 134 >> |