You may also like

More Mods

What is the units digit for the number 123^(456) ?

Expenses

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

Factorial

How many zeros are there at the end of the number which is the product of first hundred positive integers?

Vedic Sutra - All from 9 and Last from 10

Age 14 to 16 Challenge Level:

Well done Geoffrey, University College School, London for this excellent solution.

This problem is actually remarkably straightforward. A little bit of algebra is all that is needed.

Let x and y each denote 1 of the 2 numbers to be multiplied together.

The layout of the grids in the question then corresponds to:

x ( 10 - x )
y ( 10 - y )

The 4 ways of calculating the left hand digit are expressed algebraically as follows:

  1. x + y - 10
  2. 10 - [ ( 10 - x ) + ( 10 - y ) ] = 10 - ( 20 - x - y ) = x + y - 10
  3. x - ( 10 - y ) = x + y - 10
  4. y - ( 10 - x ) = x + y - 10

All 4 ways are algebraically equivalent, so this is why they give the same answer.

To explain why the Vedic Sutra works, we run through the whole method algebraically.

The right hand digit is ( 10 - x ) * ( 10 - y ).
The left hand digit is ( x + y - 10 ). We should remember that this is the tens digit and multiply by 10 before summing both sides.

10 * ( x + y - 10 ) + ( 10 - x ) * ( 10 - y )
= 10x + 10y - 100 + 100 - 10x - 10y + xy
= xy

Using the Vedic Sutra, we get the result xy, which is indeed the product of x and y.