# Nikhilam or base multiplication

This question was given in the 2^{nd} MATH-Inic Vedic Mathematics National Challenge held online last April 30, 2022, which drew 568 participants from all over the Philippines.

Anyone who has a working knowledge of of Nikhilam or base multiplication can answer this problem faster than another using an electronic calculator.

The term “Nikhilam” multiplication comes from the Vedic Maths Sutra “Nikhilam Navatashcaramam Dasatah” which means “All from 9 and the Last from 10”.

“All from 9 and the last from 10” is used to quickly find out how far the multiplicands are from the next higher powers of 10. This is fully discussed in Chapter 2 of our best selling book “25 Math Short Cuts (25 MSC) which is available online at https://www.facebook.com/MATHInicPhils).

After finding the deficiencies of the multiplicands in this example, one needs only to subtract and multiply by 2. (The proof of base multiplication for two numbers below a base can be found in Chapter 18 of 25 MSC).

The deficiencies of 888,889 and 999,998 from their base of 1,000,000 are 111,111 and 2, respectively.

The answer will be in two parts. The first part is obtained by cross-subtracting the deficiency of one multiplicand from the other: 999,998 – 111,111 or 888,999 – 2. Both will give the difference 888,887.

The second part is the product of the deficiencies: 111,111 x 2 = 222,222

The complete answer is 888,887,222, 222. Please see the figure below.

Try these exercises:

Solve the following mentally using base multiplication

- 8 x 9 =
- 7 x 6 =
- 98 x 78 =
- 97 x 87 =
- 777 x 997 =
- 789 x 998 =
- 6789 x 9997 =
- 6767 x 9998 =
- 66667 x 99997 =
- 99994 x 99996 =

Answers to last week’s exercises:

- 36 x 9 = 324
- 28 x 9 = 252
- 123 x 9 = 1107
- 136 x 9 = 1224
- 258 x 9 = 2322
- 236 x 9 = 2124
- 1456 x 9 = 13,104
- 2569 x 9 = 23,121
- 12378 x 9 = 111,402
- 14679 x 9 = 132,111