difference between the sum of its odd-placed digits and the sum of its even-placed digits is divisible by 11.

You are mistaking even and odd placed digits with even and odd digits.

Even and odd placed digits indicate tenth, thousandth, ... and units, hundreth places respectively.

Consider your example of 5181 here 5,8 are evenly placed and 1,1 are oddly place digits.

Give a thought for 827 as well, you would get the concept.

Hope it helps.

Great explanation, bubbliiiiii.

My take on this for nkaur and anyone else - that "divisibility by 11" technique is a much better parlor trick than it is a GMAT strategy. It's pretty cumbersome to apply and fairly easy to botch (I've seen people who swear they know the trick totally blow a remainder question when the difference is negative...I'll demonstrate below). And it only works for 11, so the process of memorizing it and getting good at it doesn't add a ton of value.

I'd highly recommend using a "universal divisibility strategy". Probably the quickest and most intuitive way to test a large number for divisibility by a smaller number (other than 2, 3, and 5, which have pretty good rules to them for quick analysis) is to:

-Using the fact that a(b + c) = ab + ac, subtract out known multiples of the smaller number from the larger number. For example, with 143, we know that 121 is a multiple of 11 (it's 11^2), so we could subtract 143 - 121 to get 22. And we know that 22 is 11*2, so we can break 143 into 121 + 22, or 11*11 + 11*2, or 11(11 + 2) = 11(13).

-Now, because we're just testing to see if it is, indeed, divisible, we don't even need to worry about the quotient, so we can just keep subtracting multiples of 11 (or whatever number we're testing for) until we either get to 0 (it is divisible) or we know that we won't (it's not divisible). So, trying 827, we could do:

827 - 770 (which we know to be 77 * 10, and 77 is a multiple of 11)
--> 57
57 - 55 (which we know is 5*11)
--> 2

Since 2 is what's left, we know that 827 is not divisible by 11 (we can't break 2 down any further) and we know that the remainder is 2.










Now, that 11 trick...here's where I've seen people screw it up (and adamantly so...I had a student once who stopped class and demanded control of the marker/whiteboard to prove that I was wrong...once he was done I showed him why I was right). For the question "what is the remainder when 13,331 is divided by 11", he took the alternating digits (and others have done this too):

1 + 3 + 1 = 5
3 + 3 = 6

5 - 6 = -1, so he claimed that the remainder is 1. But that's not true! The negative sign cannot be ignored. If you use the universal strategy:

13,331 - 11,000 = 2331
2331 - 2200 = 131
131 - 121 = 10

So 10 is the remainder.

My point here - don't get too fascinated with single-use tricks, because if you don't know the logic behind them you're only hope is that the GMAT tests that concept in exactly the only way you know how to use it. And if I were a GMAT question author (attention GMAT: I'm interested!), I'd write questions specifically to sniff out those one-trick-ponies and to reward those who can really think.