Seven digit base eight positive integer puzzle

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The discussion focuses on finding a seven-digit base-8 integer, N, that uses each of the digits 1 to 7 exactly once while meeting specific divisibility conditions. The conditions require that N is divisible by 7, the first six digits by 6, the first five by 5, the first four by 4, the first three by 3, and the first two by 2. One participant shares their experience of solving the problem by hand, noting that while they considered using a Mathematica script, they opted for a manual approach aided by Excel to calculate octal values. They mention that there are only 5040 permutations of the digits, which allowed them to quickly narrow down the possibilities to 48 through logical reasoning. Ultimately, they processed 36 of these possibilities to find the valid configurations for N.
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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.
 
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I'm sure I could write a mathematica script to solve this, but does anyone have any good tricks to solve it by hand?
 
NeoDevin said:
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.

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DaveE
 

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