Given Q queries which consist of two integers, one is number(1 <= number <= 106) and the other is N., the task is to find the N-th prime factor of the given number.
Input: Number of Queries, Q = 4
number = 6, N = 1
number = 210, N = 3
number = 210, N = 2
number = 60, N = 2
6 has prime factors 2 and 3.
210 has prime factors 2, 3 and 6.
60 has prime factors 2 and 3.
A naive approach is to factorize every number and store the prime factors. Print the N-th prime factors thus stored.
Time Complexity: O(log(n)) per query.
An efficient approach is to pre-calculate all the prime factors of the number and store the numbers in a sorted order in a 2-D vector. Since the number will not be more than 106, the number of unique prime factors will be around 7-8 at max(because of 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 >= 106). Once the numbers are stored, the query can be answered in O(1) as the n-1th index will have the answer in numberth row.
Below is the implementation of the above approach:
2 5 3 3
Time Complexity: O(1) per query and O(maxN * log(maxN)) for pre-processing, where maxN = 106.
Auxiliary Space: O(N * 8) in worst case
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