288. What whole number should the number 999999999 be multiplied by to get a number consisting of all ones.
288. We have $999999999=10^{9}-1,\left(10^{9}-1\right) x=\frac{10^{k}-1}{9}$; $x=\frac{10^{k}-1}{9\left(10^{9}-1\right)}$.

To make $x$ an integer, it is sufficient to take $k=81$; indeed, $x=\frac{10^{81}-1}{9\left(10^{9}-1\right)}=\frac{\left(10^{9}\right)^{9}-1}{9\left(10^{9}-1\right)}=\frac{\left(10^{9}\right)^{8}+\left(10^{9}\right)^{7}+\ldots+1}{9}$; since the sum of the digits of the numerator is 9, $x$ is an integer.