# 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:

- If the digit sum has more than one digit, we could add again until a single digit is reached.
- The digit sum is the remainder when the number is divided by 9. If the digit sum is 9, the remainder is zero.
- 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.
- 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 **2 ^{nd} 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.**