Seven digit base eight positive integer puzzle

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SUMMARY

The discussion focuses on finding a seven-digit base-8 integer, N, represented as ABCDEFG, which uses each nonzero base-8 digit (1 to 7) exactly once and meets specific divisibility conditions. The integer must be divisible by 7, 6, 5, 4, 3, and 2 for progressively smaller segments of its digits. A participant successfully narrowed down the possible combinations from 5040 permutations to 48 viable candidates using logical reasoning and Excel for calculations, demonstrating an effective manual approach to solving the puzzle.

PREREQUISITES
  • Understanding of base-8 numeral system
  • Knowledge of divisibility rules for integers
  • Familiarity with permutations and combinations
  • Basic proficiency in using Excel for calculations
NEXT STEPS
  • Explore advanced techniques in combinatorial logic
  • Learn about divisibility rules in different numeral systems
  • Investigate the use of Mathematica for solving combinatorial problems
  • Study algorithms for generating permutations efficiently
USEFUL FOR

This discussion is beneficial for mathematicians, puzzle enthusiasts, and educators interested in combinatorial logic and number theory, particularly those looking to enhance their problem-solving skills in base systems.

K Sengupta
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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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