DFA based division

Deterministic Finite Automaton(DFA) is used to check whether a number is divisible by another number k or not. If it is not divisible, then this algorithm will also find the remainder.

For the DFA based division, at first, we have to find the transition table of the DFA, using that table, we can easily find the answer. In the DFA, each state has only two transition 0 and 1.

Input and Output

Input:
The number: 50 and the divisor 3
Output:
50 is not divisible by 3 and remainder is: 2

Algorithm

dfaDivision(num, k)

Input: A number num, and divisor k.

Output: Check divisibility and the remainder.

Begin
   create transition table of size k * 2 //2 for transition 0 and 1
   state = 0
   checkState(num, state, table)
   return state
End

checkState(num, state, table)

Input: A number num, state, and the transition table.

Output: Update the state after performing division.

Begin
   if num ≠ 0, then
      tempNum := right shift number for i bit
      checkState(tempNum, state, table)
      index := number AND 1 //perform logical and with number and 1
      state := table[state][index]
End

Example

#include <iostream>
using namespace std;

void makeTransTable(int n, int transTable[][2]) {
   int zeroTrans, oneTrans;

   for (int state=0; state<n; ++state) {
      zeroTrans = state<<1;   //next state for bit 0
      transTable[state][0] = (zeroTrans < n)? zeroTrans: zeroTrans-n;

      oneTrans = (state<<1) + 1;    //next state for bit 1
      transTable[state][1] = (oneTrans < n)? oneTrans: oneTrans-n;
   }
}

void checkState(int num, int &state, int Table[][2]) {
   if (num != 0) {    //shift number from right to left until 0
      checkState(num>>1, state, Table);
      state = Table[state][num&1];
   }
}

int isDivisible (int num, int k) {
   int table[k][2];    //create transition table
   makeTransTable(k, table);    //fill the table
   int state = 0;    //initially control in 0 state
   checkState(num, state, table);
   return state;    //final and initial state must be same
}

int main() {
   int num;
   int k;
   cout << "Enter Number, and Divisor: "; cin >> num>> k;
   int rem = isDivisible (num, k);
   if (rem == 0)
      cout<<num<<" is divisible by "<<k;
   else
      cout<<num<<" is not divisible by "<<k<<" and remainder is: " << rem;
}

Output

Enter Number, and Divisor: 50 3
50 is not divisible by 3 and remainder is: 2