# 30MSC 2023 #4: Digit Sums.

Digit sum, as the term implies, is the sum of the digits of a number. Filipino students were taught early during their elementary days the divisibility rule for 9: if the sum of the digits of a number is divisible by 9, then the number is divisible by 9.

However, most of them are not taught the following very useful facts about digit sums:

1. If the digit sum has more than one digit, we could add again until a single digit is reached.
2. The digit sum is the remainder when the number is divided by 9. If the digit sum is 9, the remainder is zero.
3. In getting the digit sums, we can cast out or disregard any 9s or any digits adding up to 9 or a multiple of 9.
4. Digit sums can be used to check the results of arithmetic and polynomial operations.

Our example for today, which is a problem given in the Intermediate age group of the 2nd MATH-Inic Vedic Mathematics National Challenge held last April 2022, illustrates how digit sums can be applied to this type of arithmetic computations.

What is the remainder when the difference of 1,000,000 and 567,789 is divided by 9?

The conventional solution is, of course, to first subtract 567,789 from 1,000,000 then divide the difference by 9 to get the answer.

But we can just find the digit sums of the minuend and the subtrahend and get their difference to get answer:

For 567,789, we can cast out or disregard the ending 9 and the leading 567 ( 5 + 6 + 7 = 18). That leaves us with 7 + 8 = 15 and 1 + 5 = 6.

Now we can remove any number of 0s from 1,000,000  and still have the same digit sum. If we remove the 5 ending zeroes, we will have 10.

10 – 6 = 4, the remainder when the difference of 1,000,000 and 567,789 is divided by 9.