258. Add Digits
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Given a non-negative integer num
, repeatedly add all its digits until the result has only one digit.
For example:
Given num = 38
, the process is like: 3 + 8 = 11
, 1 + 1 = 2
. Since 2
has only one digit, return it.
Follow up:
Could you do it without any loop/recursion in O(1) runtime?
The problem, widely known as digit root problem, has a congruence formula:
https://en.wikipedia.org/wiki/Digital_root#Congruence_formula
For base b (decimal case b = 10), the digit root of an integer is:
- dr(n) = 0 if n == 0
- dr(n) = (b-1) if n != 0 and n % (b-1) == 0
- dr(n) = n mod (b-1) if n % (b-1) != 0
or
- dr(n) = 1 + (n - 1) % 9
Note here, when n = 0, since (n - 1) % 9 = -1, the return value is zero (correct).
From the formula, we can find that the result of this problem is immanently periodic, with period (b-1).
Output sequence for decimals (b = 10):
~input: 0 1 2 3 4 ...
output: 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 ....
Henceforth, we can write the following code, whose time and space complexities are both O(1).
class Solution {public: int addDigits(int num) { return 1 + (num - 1) % 9; }};
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