16 Dec
2011
16 Dec
'11
11:43 a.m.
Perhaps I'm missing something, but it seems to me that the answer to this modified question must be 1. Take the decimal log of x, and add (say) 17 to it, to get a number n, and then set N = 10^n. Now without a doubt, the smallest multiple of x exceeding 1023456789N begins with the digits 1023456789, and hence that single multiple has all ten digits in it.
x is not a fixed value; the choices of N must be constant independent of x. It's greater than 1, as (universal quantifier) N (existential quantifier) x such that Nx = 99999999 ... 99999000000 ... 00000. Sincerely, Adam P. Goucher