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162 lines
3.5 KiB
Python
Executable file
162 lines
3.5 KiB
Python
Executable file
"""#
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### 谜题描述
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Today, Osama gave Fadi an integer X, and Fadi was wondering about the minimum possible value of max(a, b) such that LCM(a, b) equals X. Both a and b should be positive integers.
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LCM(a, b) is the smallest positive integer that is divisible by both a and b. For example, LCM(6, 8) = 24, LCM(4, 12) = 12, LCM(2, 3) = 6.
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Of course, Fadi immediately knew the answer. Can you be just like Fadi and find any such pair?
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Input
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The first and only line contains an integer X (1 ≤ X ≤ 10^{12}).
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Output
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Print two positive integers, a and b, such that the value of max(a, b) is minimum possible and LCM(a, b) equals X. If there are several possible such pairs, you can print any.
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Examples
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Input
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2
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Output
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1 2
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Input
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6
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Output
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2 3
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Input
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4
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Output
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1 4
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Input
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1
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Output
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1 1
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Here is a reference code to solve this task. You can use this to help you genereate cases or validate the solution.
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```python
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def gcd(a,b):
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while b:
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a,b=b,a%b
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return a
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n=input()
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maxx=n
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for i in range(1,int(n**(0.5)+1)):
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if n%i==0:
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s=n/i
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x=gcd(i,s)
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m=max(i,s)
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m*=x
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maxx=min(maxx,m)
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a=maxx
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b=n/maxx
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b*=gcd(a,b)
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print min(a,b),max(a,b)
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```
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请完成上述谜题的训练场环境类实现，包括所有必要的方法。
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"""
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from bootcamp import Basebootcamp
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import random
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import math
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from math import gcd
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from bootcamp import Basebootcamp
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class Cfadiandlcmbootcamp(Basebootcamp):
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def __init__(self, min_X=1, max_X=10**6):
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self.min_X = min_X
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self.max_X = max_X
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def case_generator(self):
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# 增加特殊案例生成逻辑
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if random.random() < 0.3: # 30%概率生成边界案例
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candidates = [
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1, # 最小边界
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random.choice([2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31]), # 小质数
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int(math.sqrt(self.max_X))**2, # 最大平方数
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2**random.randint(1, 20), # 2的幂次方
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2 * 3 # 两个质数的乘积
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]
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X = random.choice(candidates)
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X = max(self.min_X, min(X, self.max_X))
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else:
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X = random.randint(self.min_X, self.max_X)
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return {'X': X}
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@staticmethod
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def prompt_func(question_case) -> str:
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X = question_case['X']
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prompt = f"""Given X = {X}, find two positive integers a and b such that:
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1. LCM(a, b) = X
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2. max(a, b) is minimized
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Output format: [answer]a b[/answer]"""
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return prompt
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@staticmethod
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def extract_output(output):
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import re
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matches = re.findall(r'\[answer\]\s*(\d+\s+\d+)\s*\[/answer\]', output)
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return matches[-1] if matches else None
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@classmethod
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def _verify_correction(cls, solution, identity):
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X = identity['X']
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# 解析答案
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try:
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a, b = map(int, solution.split())
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if a <= 0 or b <= 0:
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return False
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except:
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return False
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# 验证LCM正确性
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current_gcd = gcd(a, b)
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if a * b // current_gcd != X:
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return False
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# 计算最优解
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min_max = math.inf
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sqrt_x = int(math.sqrt(X))
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for i in range(1, sqrt_x + 1):
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if X % i == 0:
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j = X // i
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pair_gcd = gcd(i, j)
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candidate_max = max(i, j) * pair_gcd # 关键优化点
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min_max = min(min_max, candidate_max)
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return max(a, b) == min_max
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