Peter Winkler, in one of the first puzzles in his excellent Mathematical Puzzles: A Connoisseur's Collection, asks for proof that every positive integer can be multiplied by some (other) integer, and the resulting product will contain only the digits 0
and 1
.
As prep for a small project, I'd rather ask you: For each of the first 100 integers (or 1000 if you're ambitious), what is the smallest integer multiplicand which gives a product with only 1
and 0
among its base-10 digits?
Real question for you: How did you find them? Code welcomed, any language.
I plugged your series of solutions to this into OEIS and found
https://oeis.org/A079339
A079339 Least k such that the decimal representation of k*n contains only 1's and 0's.
which gives as its formula a relation to this sequence
A004290 Least positive multiple of n that when written in base 10 uses only 0's and 1's.
which in turn leads to
Chai Wah Wu
The American Mathematical Monthly
Vol. 121, No. 6 (June–July), pp. 529-533
which gives an existence proof via the Pigeonhole Principle.