1 billion $= 10^9$
Apply best Sorting algorithm,
$T(n) = O(n\lg n) = k.n\lg n$
The time require in seconds to sort them provided sorting thousand numbers takes 100 microseconds
$100 \times 10^{-6} = k \times 1000 \lg 1000 \\\implies k = \frac{10^{-7}}{\lg 1000}$
For n = 1 Billion let $x$ be the requied time.
$x = k \times 10^9 \lg 10^9 \\= \frac{10^{-7}}{\lg 1000} 10^9 \times \lg 10^9 \\=300 s$