Question

if p is a prime,d(N)means number divisors of N,then find the value of d(d(d(p⁵)))?

Answer 1

$$\begin{gathered} p\mathrm{is}\mathrm{a}\mathrm{prime}\mathrm{number}. \\ \mathrm{Therefore},\mathrm{divisors}\mathrm{of}{p}^5\mathrm{are}1,p,p^2,p^3,p^4\mathrm{and}{p}^5. \\ \mathrm{Number}\mathrm{of}\mathrm{divisors}\mathrm{of}{p}^5=6 \\ \therefore d\left(p^5\right)=6 \\ \mathrm{Divisors}\mathrm{of}6\mathrm{are}1,2,3\mathrm{and}6. \\ \mathrm{Number}\mathrm{of}\mathrm{divisors}\mathrm{of}6=4 \\ \therefore d\left(d\left(p^5\right)\right)=d\left(6\right)=4 \\ \mathrm{Divisors}\mathrm{of}4\mathrm{are}1,2\mathrm{and}4. \\ \mathrm{Number}\mathrm{of}\mathrm{divisors}\mathrm{of}4=3 \\ \therefore d\left(d\left(d\left(p^5\right)\right)\right)=d\left(4\right)=3. \end{gathered}$$