Maths:Not a Magic, but a Game Play

Some people do magic,don't they? We believe so because they delude us. It was about 2004. My friend got up the bench and told he wanted to show us some magic. He showed the magic as:

Ah... Take a number.
Say 135762

Add all the digits of that number :
i.e. 1+3+5+7+6+2=24

Then he told to subtract the sum from that number.
       
      135762-24=135738

Now he told us to hide a digit from that number and to give him rest of the digits. Just like
                               
      135738
      : 13538

Now he would find out what number we hid and he found out too. It was like a magic to us. But today when I remembered this thing and went through it I found it was not magic. Just he had created a kind of illusion to us.

What actually happens:

I take a number of n digits so that the number is: d1*10^n + d2*10^(n-1) + ... + d(n-1)*10 + dn*1, where d1, d2, d3,... are the digits.

Let us take the number of four digits as an example so that the number is: 1000w + 100x + 10y + z, where w, x, y, z are digits in thousand, hundred, tens and ones place.

Here the number is: 1000w + 100x + 10y + z

and the sum of digits is: w + x + y + z

What he did was subtracting the number and the sum

i.e. 1000w + 100x + 10y + z - (w + x + y + z)
   =999w + 99x + 9y
   =9(111w + 11x + y)  .............(1)

Here the result is divisible by 9 which equation (1) shows. So, the sum denoted by (1) should have the sum of its digits 9n where n is any non negative integer. But when the digit is taken off, the result becomes 9n-x. When it is divided by 9 the remainder becomes (9-x) if we take quotient as (n-1) and -x when we take quotient n. Here I don't prefer to take negative remainder so the remainder is (9-x). 'x'is the amount needed to add to (9-x) to get 9 and finally it is the hidden number.

But he did not do all these division and others. Rather he did as follows:

Let us take the previous example: 135738

Suppose I hid 7. Now he has 13538. He cancelled the set of numbers that makes sum multiples of 9 like 1,3 and 5. And then he added 3 and 8. He got 11. How much is needed to add to make it a multiple of 9?... 7. So the hidden number is 7.

This is how he showed us magic. But today when I went through it, it was a beautiful game play of mathematics, not a magic.

Hope you enjoyed it.

Thank You.