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[Hacker Rank]: Project Euler #3: Largest prime factor. Solved ✓
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# About the **Largest prime factor** solution | ||
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## Brute force method | ||
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> [!WARNING] | ||
> | ||
> The penalty of this method is that it requires a large number of iterations as | ||
> the number grows. | ||
The first solution, using the algorithm taught in school, is: | ||
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> Start by choosing a number $ i $ starting with $ 2 $ (the smallest prime number) | ||
> Test the divisibility of the number $ n $ by $ i $, next for each one: | ||
> | ||
>> - If $ n $ is divisible by $ i $, then the result is | ||
>> the new number $ n $ is reduced, while at the same time | ||
>> the largest number $i$ found is stored. | ||
>> | ||
>> - If $ n $ IS NOT divisible by $ i $, $i$ is incremented by 1 | ||
> up to $ n $. | ||
> | ||
> Finally: | ||
>> | ||
>> - If you reach the end without finding any, it is because the number $n$ | ||
>> is prime and would be the only factual prime it has. | ||
>> | ||
>> - Otherwise, then the largest number $i$ found would be the largest prime factor. | ||
## Second approach, limiting to half iterations | ||
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> [!CAUTION] | ||
> | ||
> Using some test entries, quickly broke the solution at all. So, don't use it. | ||
> This note is just to record the failed idea. | ||
Since by going through and proving the divisibility of a number $ i $ up to $ n $ | ||
there are also "remainder" numbers that are also divisible by their opposite, | ||
let's call it $ j $. | ||
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At first it seemed attractive to test numbers $ i $ up to half of $ n $ then | ||
test whether $ i $ or $ j $ are prime. 2 problems arise: | ||
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- Testing whether a number is prime could involve increasing the number of | ||
iterations since now the problem would become O(N^2) complex in the worst cases | ||
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- Discarding all $ j $ could mean discarding the correct solution. | ||
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Both problems were detected when using different sets of test inputs. | ||
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## Final solution using some optimization | ||
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> [!WARNING] | ||
> | ||
> No source was found with a mathematical proof proving that the highest prime | ||
> factor of a number n (non-prime) always lies under the limit of $ \sqrt{n} $ | ||
A solution apparently accepted in the community as an optimization of the first | ||
brute force algorithm consists of limiting the search to $ \sqrt{n} $. | ||
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Apparently it is a mathematical conjecture without proof | ||
(if it exists, please send it to me). | ||
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Found the correct result in all test cases. |
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# [Largest prime factor](https://www.hackerrank.com/contests/projecteuler/challenges/euler003) | ||
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- Difficulty: #easy | ||
- Category: #ProjectEuler+ | ||
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The prime factors of $ 13195 $ are $ 5 $, $ 7 $, $ 13 $ and $ 29 $. | ||
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What is the largest prime factor of a given number $ N $ ? | ||
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## Input Format | ||
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First line contains $ T $, the number of test cases. This is | ||
followed by $ T $ lines each containing an integer $ N $. | ||
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## Constraints | ||
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- $ 1 \leq T \leq 10 $ | ||
- $ 10 \leq N \leq 10^{12} $ | ||
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## Output Format | ||
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Print the required answer for each test case. | ||
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## Sample Input 0 | ||
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```text | ||
2 | ||
10 | ||
17 | ||
``` | ||
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## Sample Output 0 | ||
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```text | ||
5 | ||
17 | ||
``` | ||
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## Explanation 0 | ||
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- Prime factors of $ 10 $ are $ {2, 5} $, largest is $ 5 $. | ||
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- Prime factor of $ 17 $ is $ 17 $ itselft, hence largest is $ 17 $. |
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src/lib/exercises/include/exercises/hackerrank/projecteuler/euler003.hpp
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#pragma once | ||
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namespace hackerrank::projecteuler { | ||
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long euler003(long n); | ||
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} // namespace hackerrank::projecteuler |
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src/lib/exercises/src/hackerrank/projecteuler/euler003.cpp
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#include <exercises/hackerrank/projecteuler/euler003.hpp> | ||
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/** | ||
* @link Problem definition [[docs/hackerrank/projecteuler/euler003.md]] | ||
*/ | ||
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#include <cmath> | ||
#include <stdexcept> | ||
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namespace hackerrank::projecteuler { | ||
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long prime_factor(long n) { | ||
if (n < 2) { | ||
throw std::invalid_argument("n must be greater than 2"); | ||
} | ||
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long divisor = n; | ||
long max_prime_factor; | ||
bool mpf_initialized = false; | ||
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long i = 2; | ||
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while (static_cast<double>(i) <= | ||
std::sqrt(static_cast<long double>(divisor))) { | ||
if (0 == divisor % i) { | ||
divisor = divisor / i; | ||
max_prime_factor = divisor; | ||
mpf_initialized = true; | ||
} else { | ||
i += 1; | ||
} | ||
} | ||
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if (!mpf_initialized) { | ||
return n; | ||
} | ||
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return max_prime_factor; | ||
} | ||
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long euler003(long n) { return prime_factor(n); } | ||
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} // namespace hackerrank::projecteuler |
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src/tests/unit/lib/hackerrank/projecteuler/euler003.test.cpp
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#include <catch2/catch_test_macros.hpp> | ||
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#include <exercises/hackerrank/projecteuler/euler003.hpp> | ||
#include <iostream> | ||
#include <vector> | ||
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#include <filesystem> | ||
#include <fstream> | ||
#include <nlohmann/json.hpp> | ||
using json = nlohmann::json; | ||
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TEST_CASE("euler003 JSON Test Cases", | ||
"[hackerrank] [jsontestcase] [projecteuler]") { | ||
std::filesystem::path cwd = std::filesystem::current_path(); | ||
std::string path = | ||
cwd.string() + | ||
"/unit/lib/hackerrank/projecteuler/euler003.testcases.json"; | ||
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INFO("euler003 JSON test cases FILE: " << path); | ||
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std::ifstream f(path); | ||
json data = json::parse(f); | ||
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for (auto testcase : data) { | ||
long result = hackerrank::projecteuler::euler003(testcase["n"]); | ||
CHECK(result == testcase["expected"]); | ||
} | ||
} | ||
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TEST_CASE("euler003 Edge Cases", "[hackerrank] [projecteuler]") { | ||
CHECK_THROWS_AS(hackerrank::projecteuler::euler003(0), std::invalid_argument); | ||
CHECK_THROWS_AS(hackerrank::projecteuler::euler003(1), std::invalid_argument); | ||
} |
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src/tests/unit/lib/hackerrank/projecteuler/euler003.testcases.json
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[ | ||
{ "n": 10, "expected": 5 }, | ||
{ "n": 17, "expected": 17 } | ||
] |