<< problem 346 - Strong Repunits | Sum of a square and a cube - problem 348 >> |
Problem 347: Largest integer divisible by two primes
(see projecteuler.net/problem=347)
The largest integer <= 100 that is only divisible by both the primes 2 and 3 is 96, as 96=32*3=25*3.
For two distinct primes p and q let M(p,q,N) be the largest positive integer <=N only divisible by both p and q
and M(p,q,N)=0 if such a positive integer does not exist.
E.g. M(2,3,100)=96.
M(3,5,100)=75 and not 90 because 90 is divisible by 2, 3 and 5.
Also M(2,73,100)=0 because there does not exist a positive integer <= 100 that is divisible by both 2 and 73.
Let S(N) be the sum of all distinct M(p,q,N). S(100)=2262.
Find S(10 000 000).
My Algorithm
I go through all pairs (i,j) of primes i and j where i < j.
Then i analyze i^1: find the largest j^k_1 such that i^1 * j^k <= limit
In the second iteration I find the largest j^{k_2} such that i^2 * j^{k_2} <= limit
... and so on. Each time I check whether the product exceeds any previous product.
About 50% is comprised of the standard prime sieve from my toolbox.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 100 | ./347
Output:
Note: the original problem's input 10000000
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)
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>
// ---------- standard prime sieve from my toolbox ----------
// odd prime numbers are marked as "true" in a bitvector
std::vector<bool> sieve;
// return true, if x is a prime number
bool isPrime(unsigned int x)
{
// handle even numbers
if ((x & 1) == 0)
return x == 2;
// lookup for odd numbers
return sieve[x >> 1];
}
// find all prime numbers from 2 to size
void fillSieve(unsigned int size)
{
// store only odd numbers
const unsigned int half = size >> 1;
// allocate memory
sieve.resize(half, true);
// 1 is not a prime number
sieve[0] = false;
// process all relevant prime factors
for (unsigned int i = 1; 2*i*i < half; i++)
// do we have a prime factor ?
if (sieve[i])
{
// mark all its multiples as false
unsigned int current = 3*i+1;
while (current < half)
{
sieve[current] = false;
current += 2*i+1;
}
}
}
// ---------- problem-specific code ----------
int main()
{
unsigned long long limit = 10000000;
std::cin >> limit;
// compute all prime numbers
fillSieve(limit / 2);
// will contain the result
unsigned long long sum = 0;
// first prime
for (unsigned long long i = 2; i*i <= limit; i++)
{
// primes only
if (!isPrime(i))
continue;
// second prime
unsigned long long j = (i == 2) ? 3 : i + 2; // somewhat tricky way to express: "next odd number, start with 3"
for (; i*j <= limit; j += 2)
{
// primes only
if (!isPrime(j))
continue;
// largest number expressed as i^something * j^somethingelse (<= limit)
unsigned long long maxProduct = 0;
// note: j needs to be long long because otherwise a few overflows
// lead to a result which is slightly off
// i^1 * j^1
auto product = i * j;
// for i^1, i^2, i^3, ... find the maximum exponent for j
do
{
// increase exponent of j as much as possible
auto current = product;
while (current * j <= limit)
current *= j;
// better than before ?
if (maxProduct < current)
maxProduct = current;
// increment i's exponent by one
product *= i;
} while (product <= limit);
// add all maximum products
sum += maxProduct;
}
}
// display result
std::cout << sum << std::endl;
return 0;
}
This solution contains 20 empty lines, 29 comments and 2 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.07 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++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
July 17, 2017 submitted solution
July 17, 2017 added comments
Difficulty
Project Euler ranks this problem at 15% (out of 100%).
Links
projecteuler.net/thread=347 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p347.py (written by Nayuki)
Python github.com/smacke/project-euler/blob/master/python/347.py (written by Stephen Macke)
C++ github.com/Meng-Gen/ProjectEuler/blob/master/347.cc (written by Meng-Gen Tsai)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p347.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem347.java (written by Magnus Solheim Thrap)
Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own. Maybe not all linked resources produce the correct result and/or exceed time/memory limits.
Heatmap
Please click on a problem's number to open my solution to that problem:
green | solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too | |
yellow | solutions score less than 100% at Hackerrank (but still solve the original problem easily) | |
gray | problems are already solved but I haven't published my solution yet | |
blue | solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much | |
orange | problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte | |
red | problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too | |
black | problems are solved but access to the solution is blocked for a few days until a new problem is published | |
the flashing problem is the one I solved most recently |
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I scored 13,486 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
Look at my progress and performance pages to get more details.
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 !!!
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