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Thus, we can say that a number is said
to be in its generalised form if it is Quick Check
expressed as the sum of the products of 1. Write the following numbers in
its digits with their respective place generalised form:
values. (a) 55 (b) 70 (c) 121 (d) 999
2. Write the following numbers in usual form:
Games with Numbers (a) 10 × 8 + 7 (b) 100 × 7 + 10 × 7 + 7

(c) 100 × 9 + 10 × 0 + 8
Rohit is playing a game with his sister Rima.

Rima, think any
2-digit number and Alright, I choose the number
reverse the digits to 57 and its reverse is 75. So,
get a new number. 75 – 57 = 18
And subtract the
smaller number
from the bigger
number.

Now, divide the
number by 9. You will Ok, 18 ÷ 9 = 2 and
get the remainder ‘0’.
remainder is 0. Yes!
But how do you know?

Now let us learn, how Rohit finds the remainder 0 using reversing the digits of numbers.

Reversing the Digits of a 2-Digit Number and then Subtracting and Adding

The general form of the original 2-digit number ab is (10a + b). After reversing the digits, we get
the number ba = (10b + a). Now, subtract the smaller number from the larger number. That is,
• If a > b, (10a + b) – (10b + a) = 9a – 9b = 9(a – b)
• If b > a, (10b + a) – (10a + b) = 9b – 9a = 9(b – a) • If a = b, the difference will be 0.

In each case, the difference is always a multiple of 9 and when it is divided by 9, the remainder
will be ‘0’ and the quotient will always be the difference of the digits. Similarly, if we add them as
(10a + b) + (10b + a) = 11a + 11b = 11(a + b), we get the sum is always a multiple of 11 and when we
divide it by 11, the remainder will be ‘0’ and the quotient will be the sum of the digits, i.e., (a + b).

For example, let’s choose the number 13, after reversing it, we get 31. The difference between
13 and 31 is 18 (2nd multiple of 9), after dividing by 9 we get 2 as quotient and the remainder 0.
Similarly, the sum of 13 and 31 is 44, after dividing it by 11, we get the remainder 0.

339 Playing With Numbers