Even Fibonacci numbers - problem 2 >> |

# Problem 1: Multiples of 3 and 5

(see projecteuler.net/problem=1)

If we list all the natural numbers below 10 that are multiples of 3 or 5,

we get 3, 5, 6 and 9. The sum of these multiples is 23.

Find the sum of all the multiples of 3 or 5 below 1000.

# My Algorithm

We are supposed to find of all multiples of 3 or 5 *below* the input number,

therefore we decrement it by one.

In general, the sum of all numbers between 1 and x is \sum_{1..x}i=x * (x+1)/2

(see en.wikipedia.org/wiki/Triangular_number)

There are floor{x/3} numbers between 1 and x which are divisible by 3 (assuming floor{x/3} is an integer division).

e.g. the range 1..10 contains floor{10/3}=3 such numbers (it's 3, 6 and 9). Their sum is 3+6+9=18.

This can be written as 3/3 * (3+6+9) which is the same as 3 * (3/3+6/3+9/3)=3 * (1+2+3).

Those brackets represent \sum_{1..3}i = \sum_{1..10/3}i (or short: sum{10/3})

and thus our overall formula for the sum of all multiples of 3 becomes 3 * sum{x/3}.

The same formula can be used for 5:

The sum of all numbers divisible by 5 is 5 * sum{x/5}

However, there are numbers divisible by 3 *and* 5, which means they are part of *both* sums.

We must not count them twice, that's why we (in addition to the aforementioned sums)

compute the sum of all numbers divisible by 3*5=15 to correct for this error.

In the end we print `sumThree + sumFive - sumFifteen`

## Alternative Approaches

Looping through all numbers from 1 and 1000 and checking each of those numbers

whether they are divisible by 3 or 5 easily solves the problem, too, and produces the result pretty much instantly.

Even more, the code will be probably a bit shorter.

However, Hackerrank's input numbers are too large for that simple approach (up to 10^9 with 10^5 test cases)

and will lead to timeouts.

# My code

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

#include <iostream>
// triangular number: sum{x}=1+2+..+x = x*(x+1)/2

unsigned long long sum(unsigned long long x)
{
return x * (x + 1) / 2;
}
int main()
{
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned long long last;
std::cin >> last;
// not including that number
last--;
// find sum of all numbers divisible by 3 or 5
auto sumThree = 3 * sum(last / 3);
auto sumFive = 5 * sum(last / 5);
// however, those numbers divisible by 3 AND 5 will be counted twice
auto sumFifteen = 15 * sum(last / 15);
std::cout << (sumThree + sumFive - sumFifteen) << std::endl;
}
return 0;
}

This solution contains 7 empty lines, 4 comments and 1 preprocessor command.

# 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" | ./1`

Output:

*Note:* the original problem's input `1000`

__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 less than 0.01 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

February 22, 2017 submitted solution

March 23, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler001

My code solves **5** out of **5** test cases (score: **100%**)

# Difficulty

Project Euler ranks this problem at **5%** (out of 100%).

Hackerrank describes this problem as **easy**.

*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=1 - **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-problem-1/ (written by Kristian Edlund)

Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p001.hs (written by Nayuki)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p001.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p001.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/1-9/problem1.c (written by eagletmt)

Go: github.com/frrad/project-euler/blob/master/golang/Problem001.go (written by Frederick Robinson)

Javascript: github.com/dsernst/ProjectEuler/blob/master/1 Multiples of 3 and 5.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler001.scala (written by Michael Bayne)

# 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.

*Please click on a problem's number to open my solution to that problem:*

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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.

Even Fibonacci numbers - problem 2 >> |