By "abczyx" I mean 6-digit numbers in which the first three digits
are three consecutively-*in*creasing numbers, and the last
three digits are three consecutively-*de*creasing 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.)