Remainder when the difference is divided by 9.

What is the remainder when the difference between 1,000,000 and 456,789 is divided by 9?  A) 4;  B) 5;  C) 6;  D) 7;  E) 8

This question, given in the Primary category of the 3^rd MATH-Inic Vedic Mathematics National Challenge held online last April 10, 2023, can be easily solved using the most obvious method: find the difference between the two numbers, the divide the difference by 9 using the traditional long division method and find out the remainder.

But that is time consuming. Using the digit sums of the numbers will greatly shorten the computation time.

Young elementary school students are often taught how to obtain the digit sum of a number, that is, adding the digits of the number and if the total is more than 1 digit, we add again until a single digit is obtained.

As pointed out in page 3 of our book, Algebra Made Easy as Arithmetic, “What is not often emphasized is that the digit sum is the remainder when the number is divided by 9”.

The book also offers a simpler way of getting the digit sum – Casting out 9s. Using this method, we can disregard or cast out any 9s or any digits adding up to a multiple of 9.

We could then determine the remainder by

1. Using the Sutra or Vedic Math word Formula, All from 9 and the Last from 10: 1, 000, 000 – 456,789 = 543,211
2. And by Casting out 4 and 5, we are left with 3 + 2 + 1 + 1 = 7.

            An even shorter solution makes use of a form of the Sutra, The Product of the (digit) Sums is the (digit) Sum of the Product. In this case, we use the difference of the (digit) Sums is the (digit) Sum of the Difference.

            Casting out (5 + 6 + 7) and 9 will leave us with (4 + 8) = 12 and 1 + 2 = 3.

            Now we can remove 5 zeroes in 1,000,000 to get 10 – 3 = 7.