Though three digit numbers with the same three digits are in GP and AP,
once you subtract 400, the result will not have all digits the same.
So we only consider three digit numbers with all three different digits.
 

The only three digit numbers with all different digits whose digits are 
in GP that we can subtract 400 from and get a positive number
are:

421, 842, 931

421-400 = 21, not a three-digit number, so 421 is out
842-400 = 442, 4,4,2 are not in AP, so 842 is out
931-400 = 531, 5,3,1 are in AP with common difference -2.  
So that's it!

Answer: 931

Edwin
 





 

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