Question

Find the smallest number by which the given number must be multiplied so hat the product is a perfect square.
(i) 23805 (ii) 12150 (iii) 7688

Answer 1

$\left(i\right)23805=\underset{–}{3\times 3}\times 5\times 23\times 23$

Grouping the factors in pairs of equal factors,
we see that 5 is left unpaired
$\therefore$ In order to complete the pairs, we have to multiply 23805 by 5, then the product will be the perfect sqaure.
Requid smallest number = 5

(ii) $12150=2\times 3\times \underset{–}{3\times 3}\times \underset{–}{3\times 3}\times \underset{–}{5\times 5}$

Grouping the factors in pairs of equal factors,
we see that factors 2 and 3 are left unpaired
$\therefore$ In order to complete the pairs, we have to multiply 12150 by $2\times 3=6$ i.e., then the product wil be the complete sqaure.
$\therefore$ Required smallest number = 6

(iii) 7688 $=2\times \underset{–}{2\times 2}\times \underset{–}{31\times 31}$

Grouping the factors in pairs of equal factors, we see that factor 2 is left unpaired
$\therefore$ In order to complete the pairs we have to multiply 7688 by 2, then the product will be the complete square.
$\therefore$ Required smallest number = 2