Remainder with 7 for large numbers
Given a large number as a string, find the remainder of number when divided by 7.
Input : num = 1234 Output : 2 Input : num = 1232 Output : 0 Input : num = 12345 Output : 4
Simple Approach is to convert a string into number and perform the mod operation. But this approach will not work for long strings.
There exists an approach that works for large numbers also. Below are the steps.
Let the given number be “num”
- We use a series 1, 3, 2, -1, -3, -2 to find the remainder (Intuition behind the series is discussed below).
- Initialize the result as 0.
- Traverse num from end and above series from beginning. For every traversed digit, multiply it with next digit of series, and add the multiplication to result.
- Keep repeating step 3 while there are more digits to process. If there are more than 6(number of terms in series) digits, then start traversing the series from beginning.
- After each step, we keep doing result = result % 7 to make sure that result remains less than 7.
let us take above Example where number is 12345. We reverse the number from end and series from the beginning and keep adding multiplication to the result. (12345 % 7) = (5*1 + 4*3 + 3*2 + 2*(-1) + 1*(-3)) % 7 = (5 + 12 + 6 - 2 - 3)%7 = (18) % 7 = 4 hence 4 will be the remainder when we divide the number 12345 by 7.
How does this series formula work?
Below is the intuition behind the series
1 % 7 = 1 10 % 7 = 3 100 % 7 = 2 1000 % 7 = 6 = (-1) % 7 10000 % 7 = 4 = (-3) % 7 100000 % 7 = 5 = (-2) % 7 The series repeats itself for larger powers 1000000 % 7 = 1 10000000 % 7 = 3 .............. ............... The above property of modular division with 7 and associative properties of modular arithmetic are the basis of the approach used here.
Remainder with 7 is 4
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