Question

which is the smallest number by which 704 is multiplied and divided to make a perfect square?

Answer 1

$$\begin{gathered} \mathrm{Which}\mathrm{is}\mathrm{the}\mathrm{smallest}\mathrm{number}\mathrm{by}\mathrm{which}704\mathrm{is}\mathrm{multiplied}\mathrm{and}\mathrm{divided}\mathrm{to}\mathrm{make}\mathrm{a}\mathrm{perfect}\mathrm{square}? \\ \mathrm{Prime}\mathrm{factor}\mathrm{of}704=2\times 2\times 2\times 2\times 2\times 2\times 11 \\ \mathrm{So}\mathrm{here}\mathrm{we}\mathrm{can}\mathrm{see}\mathrm{that}11\mathrm{does}\mathrm{occur}\mathrm{in}\mathrm{pair}\mathrm{so}\mathrm{if}\mathrm{we}\mathrm{multiply}\mathrm{it}\mathrm{by}11\mathrm{then}\mathrm{it}\mathrm{would}\mathrm{be}\mathrm{perfect}\mathrm{square} \\ \mathrm{or}\mathrm{if}\mathrm{we}\mathrm{divide}\mathrm{it}\mathrm{by}11\mathrm{then}\mathrm{also}\mathrm{it}\mathrm{would}\mathrm{be}\mathrm{a}\mathrm{perfect}\mathrm{square}. \\ \mathrm{Perfect}\mathrm{square}\mathrm{obtain}\mathrm{by}\mathrm{dividing}11\Rightarrow 2\times 2\times 2\times 2\times 2\times 2=64 \\ \mathrm{Perfect}\mathrm{square}\mathrm{obtain}\mathrm{by}\mathrm{multiplying}11\Rightarrow 2\times 2\times 2\times 2\times 2\times 2\times 11\times 11=7744 \end{gathered}$$