Question

By what least number should the given number be multiplied to get a perfect square number? In each case, find the number whose square is the new number.
(i) $3675$
(ii) $2156$
(iii) $3332$
(iv) $2925$
(v) $9075$
(vi) $7623$
(vii) $3380$
(viii) $2475$

Answer 1

(i) Factors of 3675

$3\times 5\times 5\times 7\times 7$
$=3\times (5{)}^2\times (7{)}^2$
$\therefore$ In order to get a perfect square the given number should be multiplied by 3

The square root of the new number $=3\times 5\times 7=105$

(ii) 2156 = $2\times 2\times 7\times 7\times 11$

=$(2{)}^2\times (7{)}^2\times 11$
In order to get a perfect square, the given number should be multiplied by 11

The square root of the new number $=\left(2\right)\times \left(7\right)\times 11=154$

(iii) 3332 = $2\times 2\times 7\times 7\times 17$

= $(2{)}^2\times (7{)}^2\times 17$
In order to get a perfect square, the given number should be multiplied by 17

The square root of the new number $=\left(2\right)\times \left(7\right)\times 17=238$

(iv) 2925 = $3\times 3\times 5\times 5\times 13$

=$(3{)}^2\times (5{)}^2\times 13$
In order to get a perfect square, the given number must be multiplied by 13

The square root of the new number $=\left(3\right)\times \left(5\right)\times 13=195$

(v) 9075 = $3\times 5\times 5\times 11\times 11$

=$3\times (5{)}^2\times (11{)}^2$
In order to get a perfect square, the given number must be multiplied by 3.

The square root of the new number $=\left(3\right)\times \left(5\right)\times 11=165$

(vi) 7623 = $3\times 3\times 7\times 11\times 11$

$=(3{)}^2\times (11{)}^2\times 7$

In order to get a perfect square, the given number should be multiplied by 7

The square root of the new number $=\left(3\right)\times \left(11\right)\times 7=231$

(vii) 3380 = $2\times 2\times 5\times 13\times 13$

=$(2{)}^2\times 5\times (13{)}^2$
In order to get a perfect square, the given number must be multiplied by 5

The square root of the new number $=\left(2\right)\times \left(5\right)\times 13=130$

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

In order to get a perfect square, the given number must be multiplied by 11.

The square root of the new number $=\left(3\right)\times \left(5\right)\times 11=165$