To find the largest sum of rupees that cannot be paid using any number of coins of denominations Rs.4, Rs.8, Rs.13, and Rs.18, we can use the concept of the Frobenius coin problem.
The Frobenius coin problem states that for any two positive integers a and b that are coprime (i.e., their greatest common divisor is 1), the largest amount that cannot be paid using coins of denominations a and b is ab - a - b.
In this case, the denominations are 4, 8, 13, and 18. To find the largest sum that cannot be paid, we need to calculate the value of ab - a - b.
Let's calculate it:
ab - a - b = 4 * 8 - 4 - 8 = 32 - 4 - 8 = 20
Therefore, the largest sum of rupees that cannot be paid using coins of denominations Rs.4, Rs.8, Rs.13, and Rs.18 is 20.
Now, we need to find the sum of the digits of 20.
Sum of digits of 20 = 2 + 0 = 2
Therefore, the sum of the digits of 'X' is 2.
Hence, the correct answer is option A) 2.