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Concept: Games with Numbers

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(i) Reversing the digits – two digit number
Addition
Step-1 : Choose any 2-digit number of the form 10 x + y.
Step-2 : Reverse the digits to get a new number i.e., 
Step-3: Add the reversed number to the original number.
                (10x + y) + (10y +x) = 11x + 11y = 11(x +y)
Step-4 : Divide the answer by 11.
                11(x + y) ÷ 11 = (x + y)
Result: There is no remainder.
Remark: The sum of a two-digit number and the number formed by reversing its digits is exactly divisible by 11 and the quotient obtained is the sum of the digits of the original 2-digit number. Adding both the number, we get 36 + 63 = 99, which is exactly divisible by 11

Subtraction
Step-1 : Choose a two digit number  in the form 10x + y.
Step-2 : Reverse the digits to get a new number  in the form 10y + x.
Step-3 : Subtract both the numbers.
                (10y + x) – (10x + y) = 9y – 9x = (9 (y – x)
Step-4 : Divide the answer by 9.
                9(y – x) ÷ 9 = (y – x)
Result: There is no remainder.
Remark: The difference of a two digit number and its reversed number is exactly divisible by 9 and the quotient obtained is either the difference of the digits of the original 2-digit number or 0.

(ii) Reversing the digits – three digit number.

Addition : 
Step-1 : Choose a three digit number  xyz in the form 100x +10y + z.
Step-2 : From 2 more numbers in a way  yzx = 100z + 10x + y 
Step-3 : Add all three numbers 
(100x + 10y + z) + (100y + 10z + x) + (100z + 10x + y)
Step-4 : Divide the answer by 111. 
= 111 (x + y + z) ÷ 111 = (x + y + z).
Remark : The sum of a 3-digit number and the number formed by arranging its digits in such a way that each digit occupies a place value only once, is exactly divisible by 111

Substraction:
Step-1 : Take any three-digit number xyz in the form 100x + 10y + z. 
Step-2 : Reverse the digits : zyx = 100z + 10y + x.
Step-3 : Substract both the numbers.
 (100x + 10y + z) – (100z + 10y + x) = 99x – 99z = 99(x – z)
Remark: The difference of a 3-digit number and the number formed by reversing the digits is exactly divisible by 99.

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