Find all seven-digit numbers that contain each of the digits 0 to 6 exactly once and for which the first and last two-digit numbers are divisible by 2, the first and last three-digit numbers are divisible by 3, the first and last four-digit numbers are divisible by 4, the first and last five-digit numbers are divisible by 5, and the first and last six-digit numbers are divisible by 6.

(M. Mach)