$1\mathrm{.}31$ Here we assume that to multiply an n$-$bit number with an $m-$bit number takes time $nm.$

$1$:$31$ modified: Assume that in problem $1\mathrm{.}31$ we use the divide$-$and$-$conquer algorithm for integer multiplication $($i$.$e$.$ Karatsuba's algorithm$).$ Thus multiplying an $n$ bit number with an $m$ bit number, where $n>m,$ takes only $n^{**}{m}^{{?}^{{?}^{?}}}0\mathrm{.}585(n$ times $[m$ to the $0\mathrm{.}585$