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 \(9\).