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198 lines
7 KiB
Python
Executable file
198 lines
7 KiB
Python
Executable file
"""#
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### 谜题描述
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It is Professor R's last class of his teaching career. Every time Professor R taught a class, he gave a special problem for the students to solve. You being his favourite student, put your heart into solving it one last time.
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You are given two polynomials f(x) = a_0 + a_1x + ... + a_{n-1}x^{n-1} and g(x) = b_0 + b_1x + ... + b_{m-1}x^{m-1}, with positive integral coefficients. It is guaranteed that the cumulative GCD of the coefficients is equal to 1 for both the given polynomials. In other words, gcd(a_0, a_1, ..., a_{n-1}) = gcd(b_0, b_1, ..., b_{m-1}) = 1. Let h(x) = f(x)⋅ g(x). Suppose that h(x) = c_0 + c_1x + ... + c_{n+m-2}x^{n+m-2}.
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You are also given a prime number p. Professor R challenges you to find any t such that c_t isn't divisible by p. He guarantees you that under these conditions such t always exists. If there are several such t, output any of them.
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As the input is quite large, please use fast input reading methods.
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Input
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The first line of the input contains three integers, n, m and p (1 ≤ n, m ≤ 10^6, 2 ≤ p ≤ 10^9), — n and m are the number of terms in f(x) and g(x) respectively (one more than the degrees of the respective polynomials) and p is the given prime number.
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It is guaranteed that p is prime.
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The second line contains n integers a_0, a_1, ..., a_{n-1} (1 ≤ a_{i} ≤ 10^{9}) — a_i is the coefficient of x^{i} in f(x).
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The third line contains m integers b_0, b_1, ..., b_{m-1} (1 ≤ b_{i} ≤ 10^{9}) — b_i is the coefficient of x^{i} in g(x).
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Output
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Print a single integer t (0≤ t ≤ n+m-2) — the appropriate power of x in h(x) whose coefficient isn't divisible by the given prime p. If there are multiple powers of x that satisfy the condition, print any.
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Examples
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Input
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3 2 2
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1 1 2
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2 1
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Output
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1
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Input
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2 2 999999937
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2 1
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3 1
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Output
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2
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Note
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In the first test case, f(x) is 2x^2 + x + 1 and g(x) is x + 2, their product h(x) being 2x^3 + 5x^2 + 3x + 2, so the answer can be 1 or 2 as both 3 and 5 aren't divisible by 2.
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In the second test case, f(x) is x + 2 and g(x) is x + 3, their product h(x) being x^2 + 5x + 6, so the answer can be any of the powers as no coefficient is divisible by the given prime.
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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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n,m,p=map(int,raw_input().strip().split())
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a=map(int,raw_input().strip().split())
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b=map(int,raw_input().strip().split())
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ans=0
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for i in xrange(n):
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if a[i]%p!=0:
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ans+=i
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break
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for i in xrange(m):
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if b[i]%p!=0:
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ans+=i
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break
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print ans
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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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from bootcamp import Basebootcamp
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import re
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import random
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import math
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class Cprimitiveprimesbootcamp(Basebootcamp):
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def __init__(self, max_n=5, max_m=5, primes=None, **params):
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super().__init__(**params)
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self.max_n = max_n
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self.max_m = max_m
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self.primes = primes or [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
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def case_generator(self):
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while True:
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n = random.randint(1, self.max_n)
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m = random.randint(1, self.max_m)
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p = random.choice(self.primes)
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# Ensure first non-divisible index exists
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k_a = random.randint(0, n-1)
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k_b = random.randint(0, m-1)
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# Generate a-coefficients
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a = []
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valid_a = False
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for i in range(n):
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if i < k_a or i > k_a:
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a_val = p # Ensure other coefficients are multiples of p
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else:
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a_val = random.randint(1, 10)
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while a_val % p == 0:
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a_val = random.randint(1, 10)
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a.append(a_val)
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# Check GCD for a
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gcd_a = a[0]
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for num in a[1:]:
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gcd_a = math.gcd(gcd_a, num)
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if gcd_a == 1:
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break
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if gcd_a != 1:
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continue
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# Generate b-coefficients
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b = []
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for j in range(m):
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if j < k_b or j > k_b:
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b_val = p # Ensure other coefficients are multiples of p
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else:
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b_val = random.randint(1, 10)
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while b_val % p == 0:
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b_val = random.randint(1, 10)
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b.append(b_val)
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# Check GCD for b
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gcd_b = b[0]
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for num in b[1:]:
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gcd_b = math.gcd(gcd_b, num)
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if gcd_b == 1:
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break
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if gcd_b != 1:
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continue
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return {
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'n': n,
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'm': m,
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'p': p,
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'a': a,
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'b': b
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}
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@staticmethod
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def prompt_func(question_case):
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a_coeffs = ', '.join(map(str, question_case['a']))
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b_coeffs = ', '.join(map(str, question_case['b']))
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prompt = f"""It is Professor R's last class of his teaching career. Every time Professor R taught a class, he gave a special problem for the students to solve. You, being his favorite student, put your heart into solving it one last time.
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You are given two polynomials f(x) and g(x) with positive integral coefficients. The cumulative GCD of the coefficients of each polynomial is 1. You are also given a prime number p. Your task is to find any exponent t in the product polynomial h(x) = f(x)⋅g(x) such that the coefficient of x^t, c_t, is not divisible by p. If there are multiple such t, output any of them.
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Input format:
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- The first line contains three integers n, m, p: the number of terms in f(x), g(x), and the prime number p respectively.
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- The second line contains n integers a_0, a_1, ..., a_{{n-1}} — the coefficients of f(x).
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- The third line contains m integers b_0, b_1, ..., b_{{m-1}} — the coefficients of g(x).
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Input values for this problem:
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n = {question_case['n']}, m = {question_case['m']}, p = {question_case['p']}
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a coefficients: {a_coeffs}
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b coefficients: {b_coeffs}
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Please determine the appropriate value of t and output it. Place your final answer within [answer] and [/answer] tags. For example, if your answer is 3, write [answer]3[/answer]."""
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return prompt
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@staticmethod
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def extract_output(output):
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matches = re.findall(r'\[answer\](.*?)\[/answer\]', output, re.IGNORECASE)
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if not matches:
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return None
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return matches[-1].strip()
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@classmethod
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def _verify_correction(cls, solution, identity):
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try:
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t = int(solution)
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except:
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return False
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p = identity['p']
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a = identity['a']
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b = identity['b']
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# Find first non-divisible index in a
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i = next((idx for idx, coeff in enumerate(a) if coeff % p != 0), None)
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# Find first non-divisible index in b
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j = next((idx for idx, coeff in enumerate(b) if coeff % p != 0), None)
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return t == i + j
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