Prove: If x is a positive real number, then somewhere in the infinite sequence {x, 2x, 3x, ...} there is a number containing the digit 7.
If x is a positive real number, then somewhere in the finite sequence {x, 2x, 3x, ..., nx} there is a number containing the digit 7. Find the minimum value of n.
Note: Some numbers can be written in two ways (1.8=1.7999999...) only consider the form without all the 9's.
Source: Slightly adapted from a post by Victor Wang on Facebook.
(In reply to
re: Part I proof by Jer)
Well, then, let me put it differently. For any interval on the slide rule that you pick, I can easily name an integer that falls in that interval. That seems very obvious. For instance, the interval [log 7.32, log 7.33) includes 732, 7320, 7321, ... , 7329, 73200, 73201, etc.
That means that the set of integers hits every interval on the slide rule.
re(2): Part I proof
