Hactoberfest - Added a solution to Project Euler 120

project_euler/problem_120/sol1.py

@@ -0,0 +1,32 @@
+"""
+Problem 120 Square remainders: https://projecteuler.net/problem=120
+
+Description:
+
+Let r be the remainder when (a−1)^n + (a+1)^n is divided by a^2.
+For example, if a = 7 and n = 3, then r = 42: 6^3 + 8^3 = 728 ≡ 42 mod 49.
+And as n varies, so too will r, but for a = 7 it turns out that r_max = 42.
+For 3 ≤ a ≤ 1000, find ∑ r_max.
+
+Solution:
+
+On expanding the terms, we get 2 if n is even and 2an if n is odd.
+For maximizing the value, 2an < a*a => n <= (a - 1)/2 (integer division)
+"""
+
+
+def solution(n: int = 1000) -> int:
+    """
+    Returns ∑ r_max for 3 <= a <= n as explained above
+    >>> solution(10)
+    300
+    >>> solution(100)
+    330750
+    >>> solution(1000)
+    333082500
+    """
+    return sum(2 * a * ((a - 1) // 2) for a in range(3, n + 1))
+
+
+if __name__ == "__main__":
+    print(solution())