By "abczyx" I mean 6-digit numbers in which the first three digits are three consecutively-increasing numbers, and the last three digits are three consecutively-decreasing numbers. For example, the number 123987.

It's very easy to prove this: abczyx is the sum of aaa000, 11100, 1110, and xxx, all of which are clearly multiples of 37. (37×3 = 111.)