Question

Find a three-digit number if its digits form a geometric progression and the digits of the number which is smaller by $400$ form an arithmetic progression.

Answer 1

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 the number is 931