Question

Let $n=p_1^{{\alpha }_1},p_2^{{\alpha }_2},p_3^{{\alpha }_3}...p_k^{{\alpha }_k}$ where ($p_1,p_2,...p_k$ are primes and ${\alpha }_1,{\alpha }_2,{\alpha }_3...,{\alpha }_kϵN)$ then find the number of ways in which $n$ can be expressed as the product of two factors which are prime to one another.

Answer 1

$\begin{array}{c}x=P_1^{{\alpha }_1},P_2^{{\alpha }_2},P_3^{{\alpha }_3}.....P_k^{{\alpha }_k}\\ Lettheprimenumber\Rightarrow P_1{P}_2{P}_3.......P_k\\ n=A\times BwhereHCF\left(A,B\right)=0\end{array}$

The power of $P_1$ either A & B.

$\begin{array}{cccc}& A& B& possibleways\\ primeno.\to & 0& k& 1\\ & 1& k-1& k{c}_1\\ & 2& k-2& k{c}_2\\ & 1& & 1\\ & k& 0& k{c}_k\\ \Rightarrow \frac{1}{2}{\sum }_{r=0}^K{}^K{c}_r& \left[\begin{array}{c}\because EverycaseRepeatedbytwo\\ Timessodivideby2\end{array}\right]& & \end{array}$