Question

A number p has three different prime factors in its prime factorisation such that power of the smallest prime factor is one third the sum of the power of other two prime factors. What is the smallest value of p.

Answer 1

$$\begin{gathered} \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{question}, \\ \mathrm{p}=2^{\mathrm{x}}\times 3^{\mathrm{y}}\times 5^{\mathrm{z}}\left[\mathrm{We}\mathrm{are}\mathrm{considering}\mathrm{prime}\mathrm{factors}2,3,5\mathrm{to}\mathrm{get}\mathrm{smalles}\mathrm{p}\right] \\ \mathrm{Now}\mathrm{it}\mathrm{is}\mathrm{given}\mathrm{that} \\ \mathrm{x}=\frac{1}{3}\left(\mathrm{y}+\mathrm{z}\right) \\ \mathrm{Now}\mathrm{for}\mathrm{smalles}\mathrm{p},\mathrm{y}+\mathrm{z}=3\Rightarrow \mathrm{y}=2,\mathrm{z}=1 \\ \mathrm{So}\mathrm{x}=1 \\ \mathrm{p}=2^1\times 3^2\times 5^1=90 \end{gathered}$$