Given a large integer N, the task is to find all the possible remainders when N is divided by all the positive integers from 1 to N + 1.
Input: N = 5
Output: 0 1 2 5
5 % 1 = 0
5 % 2 = 1
5 % 3 = 2
5 % 4 = 1
5 % 5 = 0
5 % 6 = 5
Input: N = 11
Output: 0 1 2 3 5 11
Naive approach: Run a loop from 1 to N + 1 and return all the unique remainders found when dividing N by any integer from the range. But this approach is not efficient for larger values of N.
Efficient approach: It can be observed that one part of the answer will always contain numbers between 0 to ceil(sqrt(n)). It can be proven by running the naive algorithm on smaller values of N and checking the remainders obtained or by solving the equation ceil(N / k) = x or x ≤ (N / k) < x + 1 where x is one of the remainders for all integers k when N is divided by k for k from 1 to N + 1.
The solution to the above inequality is nothing but integers k from (N / (x + 1), N / x] of length N / x – N / (x + 1) = N / (x2 + x). Therefore, iterate from k = 1 to ceil(sqrt(N)) and store all the unique N % k. What if the above k is greater than ceil(sqrt(N))? They will always correspond to values 0 ≤ x < ceil(sqrt(N)). So, again start storing remainders from N / (ceil(sqrt(N)) – 1 to 0 and return the final answer with all the possible remainders.
Below is the implementation of the above approach:
0 1 2 5
- Count of distinct remainders when N is divided by all the numbers from the range [1, N]
- Find K distinct positive odd integers with sum N
- Find n positive integers that satisfy the given equations
- Find the number of positive integers less than or equal to N that have an odd number of digits
- Count of integers obtained by replacing ? in the given string that give remainder 5 when divided by 13
- Ways to write N as sum of two or more positive integers | Set-2
- Represent (2 / N) as the sum of three distinct positive integers of the form (1 / m)
- Number of ways in which N can be represented as the sum of two positive integers
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- Count of digits after concatenation of first N positive integers
- Number of arrays of size N whose elements are positive integers and sum is K
- Count positive integers with 0 as a digit and maximum 'd' digits
- Maximum number of distinct positive integers that can be used to represent N
- Check whether a number can be represented as sum of K distinct positive integers
- Permutation of first N positive integers such that prime numbers are at prime indices
- Permutation of first N positive integers such that prime numbers are at prime indices | Set 2
- Count 'd' digit positive integers with 0 as a digit
- Find the remainder when N is divided by 4 using Bitwise AND operator
- Minimum operations to make counts of remainders same in an array
- Find all the possible numbers in a range that can be evenly divided by its digits
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