Represent a number as the sum of positive numbers ending with 9
Given an integer N, the task is to check if N can be expressed as a sum of integers having 9 as the last digit (9, 19, 29, 39…), or not. If found to be true, then find the minimum count of such integers required to obtain N. Otherwise print -1.
Input: N = 156
156 = 9 + 9 + 9 + 129
Input: N = 60
No possible way to obtain sum 60 from numbers having 9 as the last digit.
Naive Approach: This problem can be viewed as a variation of the Coin change problem. For this problem, the coins can be replaced with [9, 19, 29, 39…. up to the last number smaller than N that ends with 9].
Time Complexity: O(N2)
Auxiliary Space: O(N)
Efficient Approach: The above approach can be optimized based on the observation that if the last digit of N is K, then exactly (10 – K) minimum numbers are required to form N.
A sum N can be obtained by adding 10 – K numbers, where K is the last digit of N.
Therefore, sum N can be obtained by adding 9, (9 – K) times and adding N – (9 * (9 – K)) finally.
Follow the steps below to solve the problem:
- Extract the last digit of the given number, K = N % 10
- Using the above observation, a total of (10 – K) numbers are required. Now, calculate 9 * (9 – K), as the first 9 – K numbers required to obtain N is 9.
- Now, calculate N – 9 * (9 – K) and store in a variable, say z. If z is greater than or equal to 9 and has 9 as its last digit, print 10 – K as the answer. Otherwise, print -1.
Below is the implementation of the above approach:
Time Complexity: O(1)
Auxiliary Space: O(1)