### Theory:

Playing with numbers is always an enjoyable task. We tend to figure out a few fantastic facts and these facts keep us massively entertained in the process.

Let us now observe a number game played by Alka and Yash.

Alka and Yash started to play a game. Alka asked Yash to think of any two-digit number. Once Yash has thought of one, she asked him to reverse the number. She then bet that the sum of both the two-digit numbers is a multiple of $$11$$. Yash was shocked to find it right.

Let us look at the process step-by-step:

Step $$1$$: Yash thought of $$29$$ at first.

Step $$2$$: On Alka's instructions, he then reversed the number to $$92$$.

Step $$3$$: To check Alka's statement, he added both the numbers.

$$29 + 92 = 121$$

Step $$4$$: He was then impressed to know that the total obtained is a multiple of $$11$$.

How do you think Alka could strongly bet on her statement?

It's simple logic.

There are two standard logics concerning two-digit numbers and their reverse.
Logic $$1$$: The sum of the two numbers is always a multiple of $$11$$
The general form of the two-digit number '$$ab$$' is $$(10 \times a) + b$$.

We know that to reverse a number, we should swap the numbers in TENS and ONES position.

Therefore, the general form of its reverse '$$ba$$' is $$(10 \times b) + a$$.

Let us try to find the sum of both these numbers.

$$ab + ba = [(10 \times a) + b] + [(10 \times b) + a]$$

$$= 10a + b + 10b + a$$

$$= 11a + 11b$$

$$= 11(a + b)$$

Therefore, the sum of two-digit numbers and their reverse is always a multiple of $$11$$.
Logic $$2$$: The difference between the two numbers is always a multiple of $$9$$
Let us now find the difference between the two numbers.

$$ab - ba = [(10 \times a) + b] - [(10 \times b) + a]$$

$$= 10a + b - 10b - a$$

$$= 9a - 9b$$

$$= 9(a - b)$$

Thus, it is understood that the difference between the two numbers is a multiple of $$9$$.
Example:
Let us try to apply both the logics discussed above on the number $$91$$.

Reverse of the number $$= 19$$

The sum of both these numbers $$= 91 + 19$$

$$= 110$$

$$= 11 \times 10$$

Therefore, by logic $$1$$, it is proved that the sum of two-digit numbers and their reverse is a multiple of $$11$$.

Difference between both these numbers $$= 91 -19$$

$$= 72$$

$$= 9 \times 8$$

Therefore, by logic $$2$$, it is proved that the difference between two-digit numbers and their reverse is a multiple of $$9$$.