Check whether a very large number of the given form is a multiple of 3.

Consider a very long K-digit number N with digits d₀, d₁, ..., d_K-1 (in decimal notation; d₀ is the most significant and d_K-1 the least significant digit). This number is so large that it can't be given or written down explicitly; instead, only its starting digits are given and a way to construct the remainder of the number.
Specifically, you are given d₀ and d₁; for each i ≥ 2, d_i is the sum of all preceding (more significant) digits, modulo 10, more formally - 
Determine if N is a multiple of 3.

Constraints: 
2 ≤K ≤10¹² 
1 ≤d₀ ≤9 
0 ≤d₁ ≤9 

Examples: 

Input : K = 13, d0 = 8, d1 = 1
Output : YES

Explanation: The whole number N is 8198624862486, which is divisible by 3, 
so the answer is YES. 

Input :  K = 5, d0 = 3, d1 = 4
Output : NO

Explanation: The whole number N is 34748, which is not divisible by 3, 
so the answer is NO.

Method 1 (Brute Force) 

We can apply the brute force method to calculate the whole number N by using the condition given for constructing the number iteratively (sum of preceding numbers modulo 10) and check whether the number is divisible by 3 or not. But since the number of digits (K) can be as large as 10¹², we can't store it as an integer since it will be very larger than the maximum range of 'long long int'. Hence below is an efficient method to determine if N is a multiple of 3.

Method 2 (Efficient) 

The key idea behind the solution is the fact that the digits start to repeat after some time in a cycle of length 4. Firstly, we will find the sum of all the digits and then determine if it is divisible by 3 or NOT.

We know d₀ and d₁. 
d₂ = ( d₀ + d₁ ) % 10
d₃ = ( d₂ + d₁ + d₀ ) % 10 = (( d₀ + d₁) % 10 + d₀ + d₁) % 10 = 2 * ( d₀ + d₁ ) % 10
Similarly, 
d₄ = ( d₃ + d₂ + d₁ + d₀ ) % 10 = 4 * ( d₀ + d₁ ) % 10
d₅ = ( d₄ + d₃ + d₂ + d₁ + d₀ ) % 10 = 8 * ( d₀ + d₁ ) % 10
d₆ = ( d₅ + ... + d₁ + d₀ ) % 10 = 16 * (d₀ + d₃) % 10 = 6 * ( d₀ + d₁ ) % 10
d₇ = ( d₆ + ... + d₁ + d₀ ) % 10 = 12 * ( d₀ + d₁ ) % 10 = 2 * ( d₀ + d₁ ) % 10 
 

If we keep on finding on d_i, we will see that the resultant is just looping around the same values (2, 4, 8, 6). 
Here the cycle length is 4 and d₂ is not present in the cycle. Hence after d₂ the cycle starts forming in length of 4 starting from any value in (2, 4, 8, 6) but in the same order giving a sum of S = 2 + 4 + 8 + 6 = 20 for consecutive four digits. Thus, the total sum of digits for the whole number is = d₀ + d₁ + d₂ + S*(k - 3)/4 + x, where first three terms will be covered by d₀, d₁, d₂ 
and after that groups of 4 will be covered by S. But since (k - 3) may be not a multiple of 4, some remaining digits will be left which is covered by x which can be calculated by running a loop as those number of terms will be less than 4. 

For e.g. When K = 13, 
sum of digits = d₀ + d₁ + d₂ + S * (13 - 3) / 4 + x = d₀ + d₁ + d₂ + S * 2 + x, 
where first S will have d₃, d₄, d₅, d₆ and second S will have d₇, d₈, d₉, d₁₀ and 
x = d₁₁ + d₁₂ 

• d₁₁ = 2 * ( d₀ + d₁ ) % 10
• d₁₂ = 4 * ( d₀ + d₁ ) % 10

Below is the implementation of above idea : 

// CPP Program to determine if
// number N of given form is
// divisible by 3 or not
#include <bits/stdc++.h>
using namespace std;

// function returns true if number N is
// divisible by 3 otherwise false,
// dig0 - most significant digit
// dig1 - 2nd most significant digit
// K - number of digits
bool multipleOfThree(int K, int dig0, int dig1)
{
    // sum of digits
    long long int sum = 0;

    // store the sum of first two digits
    // modulo 10 in a temporary variable
    int temp = (dig0 + dig1) % 10;

    sum = dig0 + dig1;

    // if the number N is a two digit number
    if (K == 2) {
        if (sum % 3 == 0)
            return true;
        else
            return false;
    }

    // add temp to sum to get the sum
    // of first three digits which are
    // not a part of cycle
    sum += temp;

    // get the number of groups in cycle
    long long int numberofGroups = (K - 3) / 4;

    // get the remaining number of digits
    int remNumberofDigits = (K - 3) % 4;

    // if temp = 5 or temp = 0 then sum of each group will
    // be 0
    if (temp == 5 || temp == 0)
        sum += (numberofGroups * 0);

    else
        // add sum of 20 for each group (2, 4, 8, 6)
        sum += (numberofGroups * 20);

    // find the remaining sum of remaining digits
    for (int i = 0; i < remNumberofDigits; i++) {
        temp = (2 * temp) % 10;
        sum += temp;
    }

    // check if it is divisible by 3 or not
    if (sum % 3 == 0)
        return true;
    else
        return false;
}

// Driver Code
int main()
{
    int K = 5, dig0 = 3, dig1 = 4;
    if (multipleOfThree(K, dig0, dig1))
        cout << "YES" << endl;
    else
        cout << "NO" << endl;

    K = 10;
    dig0 = 3;
    dig1 = 2;
    if (multipleOfThree(K, dig0, dig1))
        cout << "YES" << endl;
    else
        cout << "NO" << endl;
    return 0;
}

// Java Program to determine if
// number N of given form is
// divisible by 3 or not
import java.io.*;

public class GFG {

    // function returns true if number N is
    // divisible by 3 otherwise false,
    // dig0 - most significant digit
    // dig1 - 2nd most significant digit
    // K - number of digits
    static boolean multipleOfThree(int K, int dig0,
                                   int dig1)
    {

        // sum of digits
        long sum = 0;

        // store the sum of first two digits
        // modulo 10 in a temporary variable
        int temp = (dig0 + dig1) % 10;

        sum = dig0 + dig1;

        // if the number N is a two digit number
        if (K == 2) {
            if (sum % 3 == 0)
                return true;
            else
                return false;
        }

        // add temp to sum to get the sum
        // of first three digits which are
        // not a part of cycle
        sum += temp;

        // get the number of groups in cycle
        long numberofGroups = (K - 3) / 4;

        // get the remaining number of digits
        int remNumberofDigits = (K - 3) % 4;

        // add sum of 20 for each group (2, 4, 8, 6)
        sum += (numberofGroups * 20);

        // find the remaining sum of
        // remaining digits
        for (int i = 0; i < remNumberofDigits; i++) {
            temp = (2 * temp) % 10;
            sum += temp;
        }

        // check if it is divisible by 3 or not
        if (sum % 3 == 0)
            return true;
        else
            return false;
    }

    // Driver Code
    static public void main(String[] args)
    {
        int K = 5, dig0 = 3, dig1 = 4;
        if (multipleOfThree(K, dig0, dig1))
            System.out.println("Yes");
        else
            System.out.println("No");
    }
}

// This code is contributed by vt_m.

# Python 3 Program to determine if
# number N of given form is
# divisible by 3 or not

# function returns true if number N
# is divisible by 3 otherwise false,
# dig0 - most significant digit
# dig1 - 2nd most significant digit
# K - number of digits


def multipleOfThree(K, dig0, dig1):

    # sum of digits
    sum = 0

    # store the sum of first two digits
    # modulo 10 in a temporary variable
    temp = (dig0 + dig1) % 10

    sum = dig0 + dig1

    # if the number N is a
    # two digit number
    if (K == 2):
        if (sum % 3 == 0):
            return True
        else:
            return False

    # add temp to sum to get the sum
    # of first three digits which are
    # not a part of cycle
    sum += temp

    # get the number of groups in cycle
    numberofGroups = (K - 3) // 4

    # get the remaining number of digits
    remNumberofDigits = (K - 3) % 4

    # add sum of 20 for each
    # group (2, 4, 8, 6)
    sum += (numberofGroups * 20)

    # find the remaining sum of
    # remaining digits
    for i in range(remNumberofDigits):
        temp = (2 * temp) % 10
        sum += temp

    # check if it is divisible
    # by 3 or not
    if (sum % 3 == 0):
        return True
    else:
        return False


# Driver Code
if __name__ == "__main__":
    K = 5
    dig0 = 3
    dig1 = 4
    if (multipleOfThree(K, dig0, dig1)):
        print("Yes")
    else:
        print("No")

# This code is contributed by ChitraNayal

// C# Program to determine if
// number N of given form is
// divisible by 3 or not
using System;

class GFG {

    // function returns true if number N is
    // divisible by 3 otherwise false,
    // dig0 - most significant digit
    // dig1 - 2nd most significant digit
    // K - number of digits
    static bool multipleOfThree(int K, int dig0, int dig1)
    {
        // sum of digits
        long sum = 0;

        // store the sum of first two digits
        // modulo 10 in a temporary variable
        int temp = (dig0 + dig1) % 10;

        sum = dig0 + dig1;

        // if the number N is
        // a two digit number
        if (K == 2) {
            if (sum % 3 == 0)
                return true;
            else
                return false;
        }

        // add temp to sum to get the sum
        // of first three digits which are
        // not a part of cycle
        sum += temp;

        // get the number of groups in cycle
        long numberofGroups = (K - 3) / 4;

        // get the remaining number of digits
        int remNumberofDigits = (K - 3) % 4;

        // add sum of 20 for each group (2, 4, 8, 6)
        sum += (numberofGroups * 20);

        // find the remaining sum of
        // remaining digits
        for (int i = 0; i < remNumberofDigits; i++) {
            temp = (2 * temp) % 10;
            sum += temp;
        }

        // check if it is divisible by 3 or not
        if (sum % 3 == 0)
            return true;
        else
            return false;
    }

    // Driver Code
    static public void Main(String[] args)
    {
        int K = 5, dig0 = 3, dig1 = 4;
        if (multipleOfThree(K, dig0, dig1))
            Console.WriteLine("Yes");
        else
            Console.WriteLine("No");
    }
}

// This code is contributed by vt_m.

<?php
// PHP Program to determine if number N 
// of given form is divisible by 3 or not 

// function returns true if number N 
// is divisible by 3 otherwise false, 
// dig0 - most significant digit 
// dig1 - 2nd most significant digit
// K - number of digits 
function multipleOfThree($K, $dig0, $dig1)
{ 
    // sum of digits
    $sum = 0;

    // store the sum of first two digits
    // modulo 10 in a temporary variable
    $temp = ($dig0 + $dig1) % 10;

    $sum = $dig0 + $dig1;

    // if the number N is a 
    // two digit number
    if ($K == 2) 
        if ($sum % 3 == 0)
            return true;
        else
            return false;

    // add temp to sum to get the sum
    // of first three digits which are
    // not a part of cycle
    $sum += $temp;

    // get the number of groups in cycle
    $numberofGroups = (int)(($K - 3) / 4);

    // get the remaining number of digits
    $remNumberofDigits = ($K - 3) % 4;

    // add sum of 20 for each
    // group (2, 4, 8, 6)
    $sum += ($numberofGroups * 20);

    // find the remaining sum of 
    // remaining digits
    for ($i = 0; $i < $remNumberofDigits; $i++)
    {
        $temp = (2 * $temp) % 10;
        $sum += $temp;
    }

    // check if it is divisible 
    // by 3 or not
    if ($sum % 3 == 0)
        return true;
    else
        return false;
}

// Driver Code
$K = 5;
$dig0 = 3;
$dig1 = 4;
if (multipleOfThree($K, $dig0, $dig1))
    print("Yes");
else
    print("No");

// This code is contributed by mits
?>

<script>

// JavaScript Program to determine if
// number N of given form is
// divisible by 3 or not

// function returns true if number N is
// divisible by 3 otherwise false,
// dig0 - most significant digit
// dig1 - 2nd most significant digit
// K - number of digits
function multipleOfThree(K, dig0, dig1)
{
    // sum of digits
    let sum = 0;

    // store the sum of first two digits
    // modulo 10 in a temporary variable
    let temp = (dig0 + dig1) % 10;

    sum = dig0 + dig1;

    // if the number N is a two digit number
    if (K == 2) {
        if (sum % 3 == 0)
            return true;
        else
            return false;
    }

    // add temp to sum to get the sum
    // of first three digits which are
    // not a part of cycle
    sum += temp;

    // get the number of groups in cycle
    let numberofGroups = parseInt((K - 3) / 4);

    // get the remaining number of digits
    let remNumberofDigits = (K - 3) % 4;

    // if temp = 5 or temp = 0 then sum of each group will
    // be 0
    if (temp == 5 || temp == 0)
        sum += (numberofGroups * 0);

    else
        // add sum of 20 for each group (2, 4, 8, 6)
        sum += (numberofGroups * 20);

    // find the remaining sum of remaining digits
    for (let i = 0; i < remNumberofDigits; i++) {
        temp = (2 * temp) % 10;
        sum += temp;
    }

    // check if it is divisible by 3 or not
    if (sum % 3 == 0)
        return true;
    else
        return false;
}

// Driver Code
    let K = 5, dig0 = 3, dig1 = 4;
    if (multipleOfThree(K, dig0, dig1))
        document.write("YES<br>");
    else
        document.write("NO<br>");

    K = 10;
    dig0 = 3;
    dig1 = 2;
    if (multipleOfThree(K, dig0, dig1))
        document.write("YES<br>");
    else
        document.write("NO<br>");
        
</script>

NO
NO

Time Complexity: O(1)
Auxiliary Space: O(1)