253. The digits of a three-digit number form a geometric progression with different terms. If this number is decreased by 200, the result is a three-digit number whose digits form an arithmetic progression. Find the original three-digit number.
$\triangle$ The number of three-digit numbers with distinct digits forming a geometric progression is small, and all of them can be easily found by enumeration:

$124,421,139,931,248,842,469,964$.

We will discard the numbers 124 and 139, as they are less than 200. Subtract 200 from the remaining numbers and find the number (or numbers) whose digits form an arithmetic progression.

There is only one such number - 642. The original number is 842.

Answer: 842.