### 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 \(11\)

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 \(9\)

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

\(= 10a + b - 10b - a\)

\(= 9a - 9b\)

\(= 9(a - b)\)

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 \(9\).