# Digital Root

Vedic Math Olympiad Tip #4.

This question in the Intermediate category is also one of those “frequently missed” in the recent 1^{st} MATH-Inic Vedic Mathematics National Challenge. Variations of this question were also given in the younger categories and the same low percentage (25 to 30%) of correct responses were recorded.

We suspect that most contestants either do not know what a digital root is or they do not know how the Vedic Math sutra, “The Product of the Sums is the Sum of the Product” or PSSP is applied.

The digital root or the repeated digital sum of a number is obtained by adding the digits of a number. If the result has more than 1 digit, we add the digits again until a single digit remains.

We can use the technique of casting out 9s in order to shorten the determination of the digital root.

In the question given “what is the digital root of the product of 98,989 and 43,934?”, we can easily cast out the three 9s in 98,989 leaving us with two 8s. Then we can “transfer” 1 from an 8 and add it to the other 8 to get 7 and a 9: 8 + 8 = (7 + 1) + 8 = 7 + 9. The we again cast out the 9 to get a digital root of 7.

For 43,934, we can immediately cancel the 9. The we have two 3s so we need another 3 to complete a 9. We can get if from one of the 4s: 4 + 3 + 3 + 4 = 4 + 3 + 3 + (3 + 1) = 4 + 9 + 1. So if we cast out the 9, we are left with 4 + 1 or 5.

Now we do not have to multiply the two 5 digit numbers before getting the digital root of their product. We just have to use PSSP: the product of the digit sums of the numbers is equal to the digit sum of their Product.

We just have to multiply their digit sums 7 x 5 to get 35 then add 3 + 5 = 8.

We can also modify the question into “what is the remainder when the product of 98,989 and 43, 934 is divided by 9?” and the answer is still 8.

The digital root and its important applications in Arithmetic and Algebra will be discussed extensively in the 2^{nd} session of the Vedic Math Webinars Series #2 on July 25 at 3:300 to 5pm.