Question

For the following number, find the smallest whole number by which it should be divided so as to get a perfect square. Also, find the square root of the square number so obtained.
$252$
$2925$

Answer 1

By prime factorisation, we get
$252=\underset{–}{2\times 2}\times \underset{–}{3\times 3}\times 7$
Since the prime factor $7$ cannot be paired.
$\therefore$ The given number should be divided by
$\therefore \frac{252}{7}=\frac{2\times 2\times 3\times 3\times 7}{7}$
$\underset{–}{2\mid 252}\underset{–}{2\mid 126}\underset{–}{3\mid 63}\underset{–}{3\mid 21}\underset{–}{7\mid 7}\mid 1$
$=\underset{–}{2\times 2}\times \underset{–}{3\times 3}$$=36$ is a perfect square.
and, $\sqrt{36}=\sqrt{\underset{–}{2\times 2}\times \underset{–}{3\times 3}}$$=2\times 3=6$
By prime factorisation, we get
$2925=\underset{–}{3\times 3}\times \underset{–}{5\times 5}\times 13$
Since prime factor $13$ cannot be paired.
$\therefore$ The given number should be divided by $13$.
$\therefore \frac{2925}{13}=\frac{3\times 3\times 5\times 5\times 13}{13}$
$\underset{–}{3\mid 2925}\underset{–}{3\mid 975}\underset{–}{5\mid 325}\underset{–}{5\mid 65}\underset{–}{13\mid 13}\mid 1$
$=\underset{–}{3\times 3}\times \underset{–}{5\times 5}=225$ is a perfect square
and, $\sqrt{225}=\sqrt{\underset{–}{3\times 3}\times \underset{–}{5\times 5}}$$=3\times 5=15$.