 # VMO P-2 Nikhilam (Base) Multiplication: Numbers below the base

When multiplying numbers below a power of 10, we subtract one number’s deficiency from the base from the other number and then get the product of the deficiencies. Similar to base multiplication of numbers above the base, the second part of the answer has the same number of digits as the number of zeroes in the base.

In our featured example, the numbers are near both near 100. So multiplicands’ deficiencies from 100 are written to the right preceeded by minus signs.

9 8   –  2

X         9 7   –  3

98 – 3 | – 2 x – 3 = 9 5 | 0 6  = 9, 5 0 6

To get the first part of the answer,  cross subtract a deficiency from the other factor:  either (98 – 3) or (97 – 2) will give the same result which is 95.

To get the second part, multiply the deficiencies – 2 by – 3 to get 6. However, the base, 100 has two zeroes so the second part must be written as 06. The product is thus 9,506.

The proof of this method is similar to the proof shown in VMO P-1 for numbers above the base:

Let x = base

a, b  = deficiency from the base

(x – a) (x – b) = x2 – ax – bx + ab

= (x – a – b)x + ab

=  [(x – a) – b]x + ab

Let us have some more examples:

7 x 8 = 7 -2 or 8 -3 | -2 x -3 = 5|6 = 56

6 x 7 = 6 -3 or 7-4 | -3 x -4 = 3 | 12 = 42. (The base is 10 with only 1 zero, so only one place is allowed for the second part. The 1 of 12 must be carried to the next column.

89 x 87 = 89 – 13 | – 11 x – 13 = 76| 143 = 7,743 (The base is 100 with 2 zeroes, so only two places are allotted for the second part of the answer. The 1 of 143 must be carried to the next column on the left.

With a little practice, this method of multiplication can easily be done mentally.

Exercise P-2 Find the following products:

1. 6 x 9; 2) 99 x 98; 3) 98 x 93; 4) 88 x 98; 5) 75 x 97; 6) 87 x 88; 7) 97 x 67;

8) 94 x 91; 9) 995 x 975; 10) 997 x 778

Answers to Exercise J-1 from the previous post:

1. 196; 2) 324; 3) 11, 236; 4) 11, 881; 5) 12, 544; 6) 441; 7) 576; 8) 12,321;

9)  13,456; 10) 1, 002,001

• From Prudente, Virgilio, 25 Math Short Cuts, pp 65 -66.