Add euler totient function (#882)

* Add Euler's totient function implementation - Implements φ(n) using prime factorization method
- Includes comprehensive tests for small numbers, primes, prime powers, and larger values
- All tests pass and follows project naming conventions

* add to DIRECTORY.md

* Add parameterized tests for Euler totient with 100% coverage

* Add parameterized tests for Euler totient with 100% coverage

* Add parameterized tests for Euler totient with 100% coverage

* Add parameterized tests for Euler totient with 100% coverage / after echo

* code syntaxe fixing

* run cargo clippy and cargo fmt

* re-test and make sure branch is up to date

* Update src/number_theory/euler_totient.rs

Co-authored-by: Piotr Idzik <65706193+vil02@users.noreply.github.com>

* Update src/number_theory/euler_totient.rs

Co-authored-by: Piotr Idzik <65706193+vil02@users.noreply.github.com>

* add all remainning cases

* add suggestion and add other cases

* Update euler_totient.rs

Co-authored-by: Piotr Idzik <65706193+vil02@users.noreply.github.com>

* before merge

* re-test

---------

Co-authored-by: Piotr Idzik <65706193+vil02@users.noreply.github.com>

Author: triuyen

DIRECTORY.md

@@ -262,6 +262,7 @@
     * [Haversine](https://github.com/TheAlgorithms/Rust/blob/master/src/navigation/haversine.rs)
   * Number Theory
     * [Compute Totient](https://github.com/TheAlgorithms/Rust/blob/master/src/number_theory/compute_totient.rs)
+    * [Euler Totient](https://github.com/TheAlgorithms/Rust/blob/master/src/number_theory/euler_totient.rs)
     * [Kth Factor](https://github.com/TheAlgorithms/Rust/blob/master/src/number_theory/kth_factor.rs)
   * Searching
     * [Binary Search](https://github.com/TheAlgorithms/Rust/blob/master/src/searching/binary_search.rs)

src/number_theory/euler_totient.rs

@@ -0,0 +1,74 @@
+pub fn euler_totient(n: u64) -> u64 {
+    let mut result = n;
+    let mut num = n;
+    let mut p = 2;
+
+    // Find  all prime factors and apply formula
+    while p * p <= num {
+        // Check if p is a divisor of n
+        if num % p == 0 {
+            // If yes, then it is a prime factor
+            // Apply the formula: result = result * (1 - 1/p)
+            while num % p == 0 {
+                num /= p;
+            }
+            result -= result / p;
+        }
+        p += 1;
+    }
+
+    // If num > 1, then it is a prime factor
+    if num > 1 {
+        result -= result / num;
+    }
+
+    result
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    macro_rules! test_euler_totient {
+        ($($name:ident: $test_case:expr,)*) => {
+            $(
+                #[test]
+                fn $name() {
+                    let (input, expected) = $test_case;
+                    assert_eq!(euler_totient(input), expected)
+                }
+            )*
+        };
+    }
+
+    test_euler_totient! {
+        prime_2: (2, 1),
+        prime_3: (3, 2),
+        prime_5: (5, 4),
+        prime_7: (7, 6),
+        prime_11: (11, 10),
+        prime_13: (13, 12),
+        prime_17: (17, 16),
+        prime_19: (19, 18),
+
+        composite_6: (6, 2),     // 2 * 3
+        composite_10: (10, 4),   // 2 * 5
+        composite_15: (15, 8),   // 3 * 5
+        composite_12: (12, 4),   // 2^2 * 3
+        composite_18: (18, 6),   // 2 * 3^2
+        composite_20: (20, 8),   // 2^2 * 5
+        composite_30: (30, 8),   // 2 * 3 * 5
+
+        prime_power_2_to_2: (4, 2),
+        prime_power_2_to_3: (8, 4),
+        prime_power_3_to_2: (9, 6),
+        prime_power_2_to_4: (16, 8),
+        prime_power_5_to_2: (25, 20),
+        prime_power_3_to_3: (27, 18),
+        prime_power_2_to_5: (32, 16),
+
+        // Large numbers
+        large_50: (50, 20),      // 2 * 5^2
+        large_100: (100, 40),    // 2^2 * 5^2
+        large_1000: (1000, 400), // 2^3 * 5^3
+    }
+}

src/number_theory/mod.rs

@@ -1,5 +1,7 @@
 mod compute_totient;
+mod euler_totient;
 mod kth_factor;
 
 pub use self::compute_totient::compute_totient;
+pub use self::euler_totient::euler_totient;
 pub use self::kth_factor::kth_factor;