1. The problem statement, all variables and given/known data We call a seven digit number in base eight X, whose digits are given by ##abcdefg## No digits are zero, and none of them are the same. The following divisibility rules are true: (I)The number ##ab## is divisible by 2 (II) The number ##abc## is divisible by 3 (III)The number ##abcd## is divisible by 4 (IV)The number ##abcde## is divisible by 5 (V)The number ##abcdef## is divisible by 6 (VI)X is divisible by 7 Find all solutions that satisfy these constraints. No credit will be given to brute force or computational solutions. Hint: Start from proposition one and carefully reason your way through the digits. 2. Relevant equations Divisibility of a number in base n by n-1 requires that the sum of the digits also be divisible by n-1 There's probably some other rules I'm not privy to, and google hasn't produced. 3. The attempt at a solution So I started from the beginning saying that a solution to ##ab## is just an even number with no zero or repeats. So: a can be (1,2,3,4,5,6,7) and b has to be (2,4,6) Similarly: c, e, and g can also be (1,2,3,4,5,6,7) as long as there are no repeats and f has to be an even number as well so f is (2,4,6) and d has to be 4, similarly to how being divisible by 5 in base 10 means it ends in 5 or 0 and This means a, c, e, and g can only be (1,3,5,7) Also, the order of what a, c, e, and g, as well as b, and f are is irrelevant to proposition 6, since any order will be divisible by 7. I cannot figure out how to reduce this any further.