Question

What is the least natural number `n` for which the equation `floor(10^n/x)=2018` has an integer solution?

Answer 1

`7`

Explanation:

Note that `2018` has `4` significant digits, so we would expect `10^n` to be approximately the product of `2018` and another `4` digit number.

Try:

`floor(10^7/2018) = 4955`

Then:

`floor(10^7/4955) = 2018" "` as required

Is there any smaller `n` than `7`?

`floor(10^6/495) = 2020`

`floor(10^5/49) = 2040`

`floor(10^4/4) = 2500`

So `n=7` is the smallest solution.

Answer 2

See below.

Explanation:

This equation leads to the following relationship

`2017 lt 10^n/x lt 2019`

or equivalently

`10^n/2019 < x < 10^n/2017`

now for
`n = 6->495.295 < 495.786` no solution
`n=7-> 4952.95 < 4953 < 4954 < 4955 < 4956 < 4957 < 4957.86` there are `5` solutions.

so the integer solutions are `x = { 4953,4954, 4955, 4956 ,4957}` for `n = 7`