Seven digit base eight positive integer puzzle

[K Sengupta]

N is a seven digit base-8 positive integer having the form ABCDEFG that uses each of the nonzero base-8 digits 1 to 7 exactly once, and satisfies these conditions:

(i) ABCDEFG is divisible by 7.
(ii) ABCDEF is divisible by 6.
(iii) ABCDE divisible by 5.
(iv) ABCD is divisible by 4.
(v) ABC is divisible by 3.
(vi) AB is divisible by 2.

Determine all possible value(s) that N can take.

 

[NeoDevin]

I'm sure I could write a mathematica script to solve this, but does anyone have any good tricks to solve it by hand?

 

[davee123]

does anyone have any good tricks to solve it by hand?

I effectively solved by hand (I used Excel to calculate the octal values of stuff), it wasn't too bad. There's only 5040 permutations of 1-7, and the possibilities seemed to cull themselves out pretty quickly. I was able to knock it down to 48 possibilities just using some logic-- from then on it was basically slogging through-- I effectively slogged through 36, but Excel helped.

Spoiler

3254167
5234761
5674321

DaveE