Question

For each of the following numbers, find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also find the square root of the square number so obtained.
A) 14641. B) 7688 C) 2475. D) 176

Answer 1

$\bf\textsf{Given:}$

$\mathsf{14641,7688,2475,176}$

$\bf\textsf{To find:}$

$\textsf{The smallest whole number by which it should be}$

$\textsf{multiplied so as to get a perfect square}$

$\bf\textsf{Solution:}$

$\begin{array}{r|l}11&14641\\\cline{2-2}11&1331\\\cline{2-2}11&121\\\cline{2-2}11&11\\\cline{2-2}&1\\\cline{2-2}\end{array}$

$14641=11{\times}11{\times}11{\times}11$

$14641=11^2{\times}11^2{$

$\implies\bf\sqrt{14641}=11^2=121$

$\begin{array}{r|l}2&7688\\\cline{2-2}2&3844\\\cline{2-2}2&1922\\\cline{2-2}31&961\\\cline{2-2}31&31\\\cline{2-2}&1\\\cline{2-2}\end{array}$

$7688=2^2{\times}2{\times}31^2$

$\textsf{2 does not have pair}$

$\textsf{2 should be multiplied to7688}$

$15376=2^2{\times}2^2{\times}31^2$

$\implies\bf\sqrt{15376}=2{times}2{\times}31=124$

$\begin{array}{r|l}3&2475\\\cline{2-2}3&825\\\cline{2-2}5&275\\\cline{2-2}5&55\\\cline{2-2}11&11\\\cline{2-2}&1\\\cline{2-2}\end{array}$

$2475=3^2{\times}5^2{\times}11$

$\textsf{11 does not have pair}$

$\textsf{11 should be multiplied to 2475}$

$27225=3^2{\times}5^2{\times}11^2$

$\implies\bf\sqrt{27225}=3{\times}5{\times}11=165$

$\begin{array}{r|l}2&176\\\cline{2-2}2&88\\\cline{2-2}2&44\\\cline{2-2}2&22\\\cline{2-2}11&11\\\cline{2-2}&1\\\cline{2-2}\end{array}$

$176=2^2{\times}2^2{\times}11$

$\textsf{11 does not have pair}$

$\textsf{11 should be multiplied to 2475}$

$1936=2^2{\times}2^2{\times}11^2$

$\implies\bf\sqrt{1936}=2{\times}2{\times}11=44$