The approach in the book is a bit complex, I wouldnt have gotten to it myself. But there is a slightly different approach:

I did several examples:
45 and 54 = 99
32 and 23 = 55
89 and 98 = 187

And all happen to be multiples of 11, since the Units and Tens digits happen to be the same (ie 22, 33, 44, 55, 66, etc). If the sum gets past 10, this will also work in the same way.

I hope it helps.

I will try to explain this, because is a very useful approach to face other exercises.

1) You have to consider that the two digits number have the form of 10x + y. For example, 26 is 10 x 2 + 6; 99 is 10 x 9 + 9; 67 is 6 x 10 +7.

2) In this case we have two numbers, with their digits reversed. That means that the first number is 10x+y and the second one is the inverse or 10y+x. For example if the number is 54 (5x10+4) the inverse is 45 (4x10+5).

3) The sum of the two numbers is 10x+y+10y+x or 11x+11y

4) Factorizing, 11(x+y)

5) Therefore, the number is a multiple of 11. Check the answers and look for the one who is not a multiple of 11, and then voila! 181 is not a multiple of 11, thus the correct answer is A.

Saludos.

The explanation in the book is more rigorous than the examples provided by masuarezdl, although they both help you reach the same conclusion. Here's how I see it:

Let's say that t = tens digit of M and u = units digit of M. This means that M = 10t + u. This may seem confusing for you, but let's take a few numeric examples and I'm pretty sure you'll get the picture.
Let's assume we have the number 65. What does 6 in the tens position and 5 in the units position actually means? Well, it's that 65 = 6*10 + 5*1 (and I'm pretty sure you will agree with me on this). let's take another number: 79 = 7*10 + 9*1. You can pretty much tell that the tens digit means the number of 10 (TENS digit) you find in a number and that the units digit (ONE = unit) is basically the "leftovers" from dividing our number to 10.

So what they're saying over there in the GMAT Bible is that, yes indeed, since M is a two digit number, then, considering t = tens digit and u = units digit, M = 10*t + 1*u or M = 10t + u. M "is shaped" smth like [b]tu[/b]. On the other hand, the roles are reversed with N, since it's smth like [b]ut[/b], giving us N = 10*u + 1*t or N = 10u + t.
If we do the math: M + N = 10t + u + 10u + t = 11t + 11u = 11(t + u). This means that M + N is divisible by 11, as masuarezdl has noted before. So we need to cross 181 off the list, since it is not divisible by 11.

Lets say M = xy ==> 10x+y
and N = yx = 10y+ x

M+N = 11(x+y)

So the result has to be a multiple of 11.

Other than 181 all other options are divisible by 11.

So 181 is the Answer.


Thanks
Raama