[Hacker Rank]: Project Euler #3: Largest prime factor. Solved ✓.

Author: Gonzalo Diaz

docs/hackerrank/projecteuler/euler003-solution-notes.md

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+# About the **Largest prime factor** solution
+
+## Brute force method
+
+> [!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:
+
+> 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
+
+> [!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 $.
+
+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:
+
+- 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
+
+- Discarding all $ j $ could mean discarding the correct solution.
+
+Both problems were detected when using different sets of test inputs.
+
+## Final solution using some optimization
+
+> [!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} $.
+
+Apparently it is a mathematical conjecture without proof
+(if it exists, please send it to me).
+
+Found the correct result in all test cases.

docs/hackerrank/projecteuler/euler003.md

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+# [Largest prime factor](https://www.hackerrank.com/contests/projecteuler/challenges/euler003)
+
+- Difficulty: #easy
+- Category: #ProjectEuler+
+
+The prime factors of $ 13195 $ are $ 5 $, $ 7 $, $ 13 $ and $ 29 $.
+
+What is the largest prime factor of a given number $ N $ ?
+
+## Input Format
+
+First line contains $ T $, the number of test cases. This is
+followed by $ T $ lines each containing an integer $ N $.
+
+## Constraints
+
+- $ 1 \leq T \leq 10 $
+- $ 10 \leq N \leq 10^{12} $
+
+## Output Format
+
+Print the required answer for each test case.
+
+## Sample Input 0
+
+```text
+2
+10
+17
+```
+
+## Sample Output 0
+
+```text
+5
+17
+```
+
+## Explanation 0
+
+- Prime factors of $ 10 $ are $ {2, 5} $, largest is $ 5 $.
+
+- Prime factor of $ 17 $ is $ 17 $ itselft, hence largest is $ 17 $.

src/hackerrank/projecteuler/euler003.js

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+/**
+ * @link Problem definition [[docs/hackerrank/projecteuler/euler003.md]]
+ */
+
+import { BigIntMath } from '../lib/BigIntMath.js';
+
+export function primeFactor(n) {
+  if (n < 2) {
+    throw new Error('n must be greater than 2');
+  }
+
+  let divisor = n;
+  let maxPrimeFactor = divisor;
+
+  let i = 2n;
+
+  while (i <= BigIntMath.sqrt(divisor)) {
+    if (divisor % i === 0n) {
+      divisor /= i;
+      maxPrimeFactor = divisor;
+    } else {
+      i += 1n;
+    }
+  }
+
+  return maxPrimeFactor;
+}
+
+export function euler003(n) {
+  return primeFactor(n);
+}
+
+export default { euler003 };

src/hackerrank/projecteuler/euler003.test.js

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+import { describe, expect, it } from '@jest/globals';
+import { logger as console } from '../../logger.js';
+
+import { euler003 } from './euler003.js';
+
+import TEST_CASES from './euler003.testcases.json';
+
+describe('euler003', () => {
+  it('euler003 JSON Test cases', () => {
+    expect.assertions(2);
+
+    TEST_CASES.forEach((test) => {
+      const calculated = euler003(test.n);
+      console.log(`euler003(${test.n}) solution found: ${test.expected}`);
+
+      expect(`${calculated}`).toBe(`${test.expected}`);
+    });
+  });
+
+  it('euler003 Edge case', () => {
+    expect.assertions(2);
+
+    const expectedMessage = 'n must be greater than 2';
+
+    expect(() => {
+      euler003(0);
+    }).toThrow(expectedMessage);
+
+    expect(() => {
+      euler003(1);
+    }).toThrow(expectedMessage);
+  });
+});

src/hackerrank/projecteuler/euler003.testcases.json

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+[
+  { "n": 10, "expected": 5 },
+  { "n": 17, "expected": 17 }
+]