Suppose you have n elements in the array 'a' numbered from 1 to n. It is possible to remove the i-th element of 'a' if the number a[i] and i are relatively co-prime i.e. gcd(a[i],i)=1. After an element is removed, the elements to the right are shifted to the left by one position.An array b with n integers such that 1≤b[i]≤n−i+1 is a removal sequence for the array a if it is possible to remove all elements of a, if you remove the b1-th element, then the b2-th, ..., then the bn-th element. For example, let a=[42,314]:[19,19] is a removal sequence: when you remove the 1-st element of the array, the condition gcd(42,19)=1 holds, and the array becomes [314];when you remove the 1-st element again, the condition gcd(314,19)=1 holds, and the array becomes empty.[2,1] is not a removal sequence: when you try to remove the 2-nd element, the condition gcd(314,2)=1 is false.An array is special if it has at least two removal sequences. For example, the array [1,2,5] is special: it has removal sequences [3,1,1] and [1,2,1]. The array [42,314] is not special: the only removal sequence it has is [1,1].You are given two integers n and m. You have to calculate the number of special arrays a such that the length of a is from 1 to n and each ai is an integer from 1 to m.Input FormatThe first line of the input contains two integers n and m.ConstraintsPrint one integer — the number of special arrays a such that the length of a is from 1 to n and each a[i] is an integer from 1 to m. Since the answer can be very large, print it modulo 998244353.Output Format2≤n≤3⋅10^51≤m≤10^12Sample Input 010 24Sample Output 0406957500Sample Input 12 4Sample Output 18
Question
Suppose you have n elements in the array 'a' numbered from 1 to n. It is possible to remove the i-th element of 'a' if the number a[i] and i are relatively co-prime i.e. gcd(a[i],i)=1. After an element is removed, the elements to the right are shifted to the left by one position.An array b with n integers such that 1≤b[i]≤n−i+1 is a removal sequence for the array a if it is possible to remove all elements of a, if you remove the b1-th element, then the b2-th, ..., then the bn-th element. For example, let a=[42,314]:[19,19] is a removal sequence: when you remove the 1-st element of the array, the condition gcd(42,19)=1 holds, and the array becomes [314];when you remove the 1-st element again, the condition gcd(314,19)=1 holds, and the array becomes empty.[2,1] is not a removal sequence: when you try to remove the 2-nd element, the condition gcd(314,2)=1 is false.An array is special if it has at least two removal sequences. For example, the array [1,2,5] is special: it has removal sequences [3,1,1] and [1,2,1]. The array [42,314] is not special: the only removal sequence it has is [1,1].You are given two integers n and m. You have to calculate the number of special arrays a such that the length of a is from 1 to n and each ai is an integer from 1 to m.Input FormatThe first line of the input contains two integers n and m.ConstraintsPrint one integer — the number of special arrays a such that the length of a is from 1 to n and each a[i] is an integer from 1 to m. Since the answer can be very large, print it modulo 998244353.Output Format2≤n≤3⋅10^51≤m≤10^12Sample Input 010 24Sample Output 0406957500Sample Input 12 4Sample Output 18
Solution
This problem is a competitive programming problem that requires knowledge of number theory and dynamic programming. Here is a step-by-step guide on how to solve it:
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Understand the Problem: The problem is asking for the number of special arrays of length from 1 to n where each element is an integer from 1 to m. An array is special if it has at least two removal sequences.
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Define the Constraints: The constraints are that 2≤n≤3⋅10^5 and 1≤m≤10^12. The answer can be very large, so it should be printed modulo 998244353.
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Approach: The problem can be solved using dynamic programming. We can define a 2D DP array dp[i][j] where dp[i][j] is the number of special arrays of length i with j removal sequences. We can then iterate over all possible lengths and removal sequences and update our DP array accordingly.
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Implementation: First, initialize the DP array with all zeros. Then, for each length i from 1 to n and for each removal sequence j from 1 to i, calculate dp[i][j] as the sum of dp[i-1][k] for all k from 1 to j-1, multiplied by (m-1). Then, add the product of j and m to dp[i][j]. Finally, take the modulo of dp[i][j] with 998244353 to ensure the answer does not exceed the limit.
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Output: The answer is the sum of all dp[n][j] for all j from 2 to n, modulo 998244353.
Please note that this problem requires a good understanding of dynamic programming and number theory, and the time complexity of the solution is O(n^2), which is feasible given the constraints.
Similar Questions
You are given a 0-indexed array of distinct integers nums.There is an element in nums that has the lowest value and an element that has the highest value. We call them the minimum and maximum respectively. Your goal is to remove both these elements from the array.A deletion is defined as either removing an element from the front of the array or removing an element from the back of the array.Return the minimum number of deletions it would take to remove both the minimum and maximum element from the array.
Problem StatementRajini is a student who loves coding. He is working on an array operations problem. In the given problem, Rajini needs to remove the first element from the array and print the modified array.Help Rajini write a program that takes an array of integers as input, deletes the first element from the array, and prints the modified array.Input format :The first line of input is an integer n, representing the number of elements in the array.The second line of input consists of n space-separated integers, representing the elements of the array arr[i].Output format :The output displays n space-separated integers of the modified array after deleting the first element.Refer to the sample output for the formatting specifications.Code constraints :In this scenario, the test cases will fall under the following constraints:2 ≤ n ≤ 101 ≤ arr[i] ≤ 100Sample test cases :Input 1 :56 7 4 3 1Output 1 :7 4 3 1 Input 2 :210 47Output 2 :47 Input 3 :1037 84 27 23 48 19 1 38 27 39 100Output 3 :84 27 23 48 19 1 38 27 39
When deleting an element from the middle of a 1D array, what is the primary challenge?
What is the time complexity of removing at index n in an unsorted array?
Egath is participating in a coding hackathon and one of the challenges requires him to work with an array of integers. The task is to remove exactly one element from the array such that the sum of the remaining elements is a prime number. Help Egath find the first possible prime sum by removing one element or determining if no such prime sum can be achieved.Input format :The first line of input consists of an integer N, representing the number of elements in the array.The second line consists of N space-separated integers, representing the array elements.Output format :If removing one element results in a prime sum, print the sum.If no such prime sum can be achieved by removing exactly one element, print "No valid prime sum found".Refer to the sample output for formatting specifications.Code constraints :The given test cases fall under the following constraints:1 ≤ N ≤ 151 ≤ elements of the array ≤ 1000Sample test cases :Input 1 :31 2 3Output 1 :5Input 2 :41 2 4 6Output 2 :11Input 3 :42 3 5 7Output 3 :No valid prime sum found
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