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2-digit number - reverse 2-digit number = Multiple of 9

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by XLogic » Mon Feb 21, 2011 8:22 pm
Hello,

I actually stumbled across this fact when working with some numbers... And it was later confirmed by Manhattan GMAT Strategy guide.

Any 2-digit number minus the reverse of said 2-digit number = multiple of 9 (the 2 digits must be unique of course)
E.g. 21-12 = 9 72-27 = 45 81-18 = 63 etc.

Can someone provide a quick explanation of this phenomenon. And please, no crazy math talk here-- just something simple for the feeble mind to grasp :-)

Thanks.
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by Anurag@Gurome » Mon Feb 21, 2011 8:35 pm
A simple explanation:
Let the two-digit integer be 10x+y where x and y are single digits.
On reversing, the new integer is 10y+x.
The difference is (10x+y) - (10y+x) = 9(x-y).
Since 9 is a factor of the difference, the difference has to be a multiple of 9.
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