(In reply to
my conjecture by Jer)
Lets prove this with basic induction.
Claim: If a number is represented by a string of 3^x of the digit 3, then the number is divisible by 3^(x+1).
Base case x=1. Then the number is 333. We expect 3^2 to divide the number. 333 factors into 3^2*37. 3^2 does in fact appear in the factorization of 333. Base case established.
Inductive step. Let M be the number represented by a string of 3^x of the digit 3 and similarly, N be the number represented by a string of 3^(x+1) of the digit 3.
The inductive step is to assume M is divisible by 3^(x+1), and then show N must then be divisible by 3^(x+2).
N can be expressed as M*100..00100...001. M is assumed to be a multiple of 3^(x+1). 100..00100...001 is congruent to 3 mod 9, so it is a multiple of 3 but not 9. Therefore N is a multiple of 3^(x+1)*3 = 3^(x+2). Inductive step proved.
We have established the base case and proven the induction step holds, so the entire induction holds and we have proven the claim.
re: my conjecture - a proof
