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question:Write a function `find_gcd_sequence` that takes an integer `N` and finds the longest sequence of consecutive integers starting from `1` such that the greatest common divisor (GCD) of all those integers is greater than `1`. To clarify, the sequence must start from `1` and consist of consecutive positive integers. The objective is to find the maximum possible length `L` of this sequence where the GCD of all `L` integers is greater than `1`. If there's no such sequence with length greater than `1`, return `0`. The function should return an integer representing the length of the longest sequence with the required property. # Input - The function takes a single integer `N` (1 ≤ N ≤ 10^18) which represents the upper bound on the potential length of the sequence (you don't need to consider sequences longer than `N`). # Output - Return an integer representing the length of the longest sequence starting from `1` such that the GCD of all the integers in this sequence is greater than `1`. # Example - For `N = 6`, the longest sequence is `[1, 2, 3, 4, 5, 6]`, and the GCD of these numbers is `1`, so the return value should be `0`. - For `N = 4`, the longest sequence `[1, 2, 3, 4]` also has a GCD of `1`, so the return value should be `0`. - For `N = 10`, the longest sequence `[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]` has a GCD of `1`, so the return value should be `0`. # Constraints - Since the input number `N` can be very large, the solution should consider the efficiency of calculations to avoid timeouts. # Solution Outline To solve this problem, you can use the property of GCD and prime numbers: - The GCD of a set of numbers will be greater than `1` if and only if all numbers share at least one prime number as a factor. - Therefore, the sequence with a GCD greater than `1` must consist of multiples of a prime number. The length of the sequence is maximized starting from `1` and ending before `N`. - You'll need to iteratively check sequences by their prime factors and find the maximum sequence that satisfies the condition. You may use helper functions to check for primes and calculate GCDs of sequences efficiently. Consider edge cases and scenarios where no valid sequence can be formed.
answer:def find_gcd_sequence(N): Finds the longest sequence of consecutive integers starting from 1 such that the greatest common divisor (GCD) of all those integers is greater than 1. :param N: int - upper bound on the potential length of the sequence (1 ≤ N ≤ 10^18) :return: int - the length of the longest sequence with the required property if N <= 1: return 0 # As the sequence must start from 1 and consist of consecutive positive integers, # their GCD needs to be greater than 1. # The only possibility for a GCD > 1 would mean (1, 2, 3, ..., N) must have some common factor > 1. # Since 1 causes the GCD to always be 1, the result should be 0 for all N. # But we need to find sequence of length greater than 1, and that means including multiple of primes. # GCD of consecutive integers including 1 will always be 1, so return 0. return 0
question:Palindrome Checker Write a function that takes a string as input and determines whether it is a palindrome, ignoring non-alphanumeric characters and case. The function should return `True` if the string is a palindrome and `False` otherwise. A palindrome is a word, phrase, number, or other sequences of characters that reads the same forward and backward (ignoring spaces, punctuation, and capitalization). # Examples ```python is_palindrome("A man, a plan, a canal, Panama") # should return True is_palindrome("race a car") # should return False is_palindrome("No 'x' in Nixon") # should return True is_palindrome("Was it a car or a cat I saw?") # should return True is_palindrome("Not a palindrome") # should return False ``` # Input - A single string containing the phrase or sentence to check. # Output - A boolean value: `True` if the input string is a palindrome, `False` otherwise. The function should ignore: - Punctuation - White spaces - Differences in letter case # Constraints - The input string can contain any printable ASCII characters. - The length of the string will be between 1 and 1000 characters inclusive.
answer:import re def is_palindrome(s): Determines whether the input string is a palindrome, ignoring non-alphanumeric characters and case. Parameters: s (str): The input string to check. Returns: bool: True if the string is a palindrome, False otherwise. # Use regex to filter out non-alphanumeric characters and convert to lower case filtered_string = re.sub(r'[^a-zA-Z0-9]', '', s).lower() # Check if the filtered string reads the same forward and backward return filtered_string == filtered_string[::-1]
question:Task You are given a string consisting of English alphabet letters. You need to determine if this string is a valid palindrome after removing at most one character. A palindrome is a string that reads the same forward and backward. # Example For `s = "racecar"`, the result should be `true`. `racecar` is already a palindrome. For `s = "raceca"`, the result should be `true`. Removing 'a', `raceca` becomes "racec" which is a palindrome. For `s = "abcdefg"`, the result should be `false`. No single character removal can make it a palindrome. For `s = "abca"`, the result should be `true`. Removing 'c', `abca` becomes "aba" which is a palindrome. # Input / Output - `[input]` string s Non-empty string where `1 ≤ length of s ≤ 1000` - `[output]` a boolean value `true` if the string can become a palindrome after removing at most one character, `false` otherwise.
answer:def is_valid_palindrome(s: str) -> bool: Determines if the string can be a palindrome after removing at most one character. Args: s: The input string. Returns: A boolean indicating whether the string can be a palindrome after removing at most one character. def is_palindrome_range(i: int, j: int) -> bool: while i < j: if s[i] != s[j]: return False i += 1 j -= 1 return True left, right = 0, len(s) - 1 while left < right: if s[left] != s[right]: return is_palindrome_range(left + 1, right) or is_palindrome_range(left, right - 1) left += 1 right -= 1 return True
question:In a remote village, the electric grid is organized in a circular manner. There are n houses connected in a circle, and each house is supplied with power through an individual transformer. Each transformer has a certain power rating represented by an integer (in kW). The power output of each transformer can be increased or decreased, but changing a transformer's power rating consumes resources. The village's council has decided that the power ratings of the transformers must follow a specific pattern for better efficiency: the power rating of the i-th transformer should be equal to the power rating of the (i + n/2)-th transformer, where indices are considered modulo n. Importantly, if n is odd, only every other transformer should be adjusted to follow this pattern since perfect symmetry is impossible. Determine the minimum number of transformers that need to be changed to achieve the desired pattern. # Input - The first line contains an integer n (1 ≤ n ≤ 100,000), the number of houses in the village. - The second line contains n integers ai (1 ≤ ai ≤ 100,000), the power ratings of the transformers. # Output - Print a single integer, the minimal number of transformers that need to be adjusted. # Examples Input ``` 6 3 4 5 3 4 5 ``` Output ``` 0 ``` Input ``` 5 1 2 3 4 5 ``` Output ``` 2 ```
answer:def min_transformers_to_adjust(n, power_ratings): Determines the minimum number of transformers that need to be adjusted to achieve the desired pattern. Parameters: n (int): The number of houses in the village. power_ratings (list of int): The power ratings of the transformers. Returns: int: The minimum number of transformers that need to be adjusted. if n == 1: return 0 count = 0 half_n = n // 2 for i in range(half_n): if power_ratings[i] != power_ratings[(i + half_n) % n]: count += 1 return count # Sample usage # n = 6 # power_ratings = [3, 4, 5, 3, 4, 5] # print(min_transformers_to_adjust(n, power_ratings)) # Output: 0