Does every positive integer have a multiple whose decimal representation consists only of 0's and 1's?
arithmetic/digits/zeros.and.ones New
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Problem
Does every positive integer have a multiple whose decimal representation consists only of 0's and 1's?
Show solution
Yes. Let n be a positive integer. Write n=a*b, where a is divisible by
no primes other than 2 or 5, and gcd(b,10) = 1.
a divides 10^k for some k.
Let r = the number of integers in the set {1,...,b-1} which are
relatively prime to b. By Euler's theorem, 10^r = 1 (mod b).
Now, 1 + 10^r + 10^(2r) + . . . + 10^((b-1)r) = 0 (mod b).
Therefore, 10^k [1 + 10^r + 10^(2r) + . . . + 10^((b-1)r)] is divisible
by n.
-- David Radcliffe <radcliff@csd4.csd.uwm.edu>
[ Editor: see also entry "all.ones". ]