Question

For each of the following number, find the smallest number by which it should be multipled so as to get a perfect square. Also find the square root of the square number obtained

i) 180

ii) 1458 [4 MARKS]

Answer 1

Prime factorisation : 1 Mark each
Conclusion : 1 Mark each

i) Prime factorisation of 180

$\begin{array}{cc}2& 180\\ 2& 90\\ 3& 45\\ 3& 15\\ 5& 5\\ & 1\end{array}$

$\therefore 180=2\times 2\times 3\times 3\times 5$

If we multiply 180 by 5, we will get $\underset{–}{2\times 2}\times \underset{–}{3\times 3}\times \underset{–}{5\times 5}$

$\therefore \sqrt{900}=2\times 3\times 5=30$

ii) Prime factorization of 1458 is :

$\begin{array}{cc}2& 1458\\ 3& 729\\ 3& 243\\ 3& 81\\ 3& 27\\ 3& 9\\ 3& 3\\ & 1\end{array}$

$1458=2\times 3\times 3\times 3\times 3\times 3\times 3$

Clearly, we can see that prime factor 2 is unpaired therefore, we need to multiply 1458 by 2 to get a perfect square

$\therefore 1458\times 2=2916=\underset{–}{2\times 2}\times \underset{–}{3\times 3}\times \underset{–}{3\times 3}\times \underset{–}{3\times 3}$, and

$\sqrt{2916}=2\times 3\times 3\times 3=54$