Magic number
Number Theory
Level
2
Define a number magical if the number are reversed, the absolute difference are divisible by 9.
How many 2 digit numbers that are not magical?? ( A Magical number doesn't end with 0)
69 is magical because 69-96 = -27, 27 is divisible by 9
The answer is 9.
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Note that all 2 digit numbers can be written as 1 0 a + b where a and b are the digits. Thus, we have
( 1 0 a + b ) − ( 1 0 b + a ) = 9 a − 9 b = 9 ( a − b )
Clearly, this is divisible by 9, which implies that all 2 digit numbers are magical. But sine magical numbers do not end in 0, there are 9 numbers which are not magical. Therefore, the answer is 9.