88 lines
1.9 KiB
Python
88 lines
1.9 KiB
Python
"""
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2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.
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What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?
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"""
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import math
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import random
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import time
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def timer(func):
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def wrapper(*args, **kwargs):
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start_time = time.time()
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result = func(*args, **kwargs)
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end_time = time.time()
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print(f"{func.__name__} took {end_time - start_time:.6f} seconds")
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return result
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return wrapper
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def is_probable_prime(n: int, trials: int = 10) -> bool:
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"""Miller-Rabin素性测试（快速判断是否为质数）"""
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if n < 2:
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return False
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if n in (2, 3):
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return True
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if n % 2 == 0:
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return False
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# 将 n-1 写成 d * 2^s 的形式
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d = n - 1
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s = 0
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while d % 2 == 0:
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d //= 2
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s += 1
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# 测试
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for _ in range(trials):
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a = random.randrange(2, n - 1)
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x = pow(a, d, n)
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if x == 1 or x == n - 1:
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continue
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for _ in range(s - 1):
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x = pow(x, 2, n)
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if x == n - 1:
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break
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else:
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return False
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return True
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def primes_up_to(n: int) -> list[int]:
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if n < 2:
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return []
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if n == 2:
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return [2]
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if n == 3:
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return [2, 3]
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primes = [2, 3]
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for i in range(5, n + 1, 2):
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if is_probable_prime(i):
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primes.append(i)
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return primes
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@timer
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def smallest_multiple(n: int) -> int:
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result = 1
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for p in primes_up_to(n):
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# 找出最大幂次
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exp = int(math.log(n, p))
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result *= p**exp
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return result
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@timer
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def python_solution(n: int) -> int:
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ls = list(range(1, n + 1))
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return math.lcm(*ls)
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if __name__ == "__main__":
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print(smallest_multiple(20))
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print(python_solution(20))
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