Problem Statement:
A 2-digit number is reversed. The larger of the two numbers is divided by the smaller one. What is the largest possible remainder?

Solution:
Let's assume the original 2-digit number is 10x + y, where x and y are digits of the number.
If we reverse the number, we get 10y + x.
The larger of the two numbers is the one with the higher value, which is max(10x + y, 10y + x).
The smaller of the two numbers is the one with the lower value, which is min(10x + y, 10y + x).

Now, we need to find the largest possible remainder when the larger number is divided by the smaller number.
Let's assume that the larger number is (10x + y) and the smaller number is (10y + x).

The remainder when (10x + y) is divided by (10y + x) can be found using the following formula:

(10x + y) mod (10y + x) = (10x + y) - k(10y + x)

where k is an integer such that (10x + y) - k(10y + x) is as small as possible but still greater than or equal to zero.

Simplifying the above equation, we get:

(10x + y) mod (10y + x) = (9kx - ky) mod (10y + x)

Since we want to find the largest possible remainder, we need to find the largest possible value of (9kx - ky) that is less than or equal to (10y + x).

We can start by assuming k = 1 and finding the largest possible value of (9x - y) that is less than or equal to (10y + x). If this value is negative, we can increase k by 1 and repeat the process until we find a positive value.

Let's try this approach for each possible value of y from 1 to 9 and see which value gives us the largest remainder.

y = 1: (9x - 1) <= 10="" +="" x="" --=""> 8x <= 11="" --=""> x = 1 or x = 2
If x = 1, then (10x + y) = 11 and (10y + x) = 21, so the remainder is 11.
If x = 2, then (10x + y) = 21 and (10y + x) = 12, so the remainder is 9.

y = 2: (9x - 2) <= 20="" +="" x="" --=""> 8x <= 22="" --=""> x = 2 or x = 3
If x = 2, then (10x + y) = 22 and (10y + x) = 32, so the remainder is 22.
If x = 3, then (10x + y) = 32 and (10y + x) = 23, so the remainder is 15.

y = 3: (9x - 3) <= 30="" +="" x="" --=""> 8x <= 33="" --=""> x = 4 or x = 5
If x = 4, then (10x + y) = 43 and (10y + x) = 34, so the remainder