Question 44
For a positive integer $d$, let $f(d)$ be the smallest positive integer with exactly $d$ positive divisors. (E.g. $f(1) = 1$, $f(5) = 16$, $f(6) = 12$.)
Prove that for any positive integer $k$ the number $f(2^{k-1})$ divides $f(2^k)$.
Prove that for any positive integer $k$ the number $f(2^{k-1})$ divides $f(2^k)$.