# The Difference of the Sums is the Sum of the Difference

This question was given in the Primary category in the 2^{nd} MATH-Inic Vedic Mathematics National challenge held online last April 30, 2022.

The conventional approach to this problem is of course, 1) perform the subtraction and 2) divide the difference by 9 to get the answer.

But by using a form of the Vedic Math Sutra, “the Product of the Sums is the Sum of the Product (PSSP)” we can get the answer in less than 3 seconds!

Digit sums or digital root and PSSP are discussed in detail in our book “Algebra Made Easy as Arithmetic” which is available at https://www.facebook.com/MATHInicPhils

PSSP is used mainly to check the results of arithmetic operations. Here the applicable form is “The Difference of the Sums is the Sum of the Difference”. This means that difference between the digit sum of the minuend and the digit sum of the subtrahend is equal to the digit sum of the difference.

But since we also know that the digital root or repeated digital sum is the remainder if a number is divided by 9, we can reduce the numbers into their digit sums before performing the subtraction.

Determining the digit sum is further simplified by “casting out 9s” – or removing or disregarding any 9s or any number combinations adding up to 9 or a multiple of 9. In our problem, we can readily cast out the 9 and “5 + 6 + 7” in 56,789 leaving a digit sum of 8.

For the minuend we can disregard the 4 ending zeroes which will result into a simple “10 – 8” subtraction. We can easily verify the “2” is indeed the remainder when the difference is divided by 9.

Questions such as “what is the remainder when the product of 6789 x 9876 is divided by 9” or “what is the digital root of the sum of 98765 and 88776” can be easily solved using PSSP and its different forms.

Practice exercises: Find the remainder when the results of the following operations are divided by 9.

- 327 + 633 =
- 989 – 543 =
- 789 x 536 =
- 22712 ÷ 334 =
- 7788 + 8877 =
- 8641 – 6421 =
- 3468 x 5672 =
- 24375 + 89731 =
- 76534 – 18939 =
- 77777 x 88888 =

Answers to last week’s exercises:

- 8 x 9 = 72
- 7 x 6 = 42
- 98 x 78 = 7644
- 97 x 87 = 8439
- 777 x 997 =774,669
- 789 x 998 =787,422
- 6789 x 9997 = 67,869,633
- 6767 x 9998 =67,656,466
- 66667 x 99997 = 6,666,499,999
- 99994 x 99996 = 9,999,000,024