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Find the smallest number by which the given number must be multiplied so that the product is a perfect square.
(i) 23805
(ii) 12150
(iii) 7688.
To do :
We have to find the smallest numbers by which the given numbers must be multiplied so that the products are perfect square.
Solution:
Perfect Square: A perfect square has each distinct prime factor occurring an even number of times.
(i) Prime factorisation of 23805 $=3\times3\times5\times23\times23$
$=(3)^2\times5\times(23)^2$
$=(3\times23)^2\times5$
$=(69)^2\times5$
In order to make the pairs an even number of pairs, we have to multiply 23805 by 5, then the product will be the perfect square.
Therefore, 5 is the smallest number by which 23805 must be multiplied so that the product is a perfect square.
(ii) Prime factorisation of 12150 $=2\times3\times3\times3\times3\times3\times5\times5$
$=2\times3\times(3)^2\times(3)^2\times(5)^2$
In order to make the pairs an even number of pairs, we have to multiply 12150 by $2\times3=6$, then the product will be the perfect square.
Therefore, 6 is the smallest number by which 12150 must be multiplied so that the product is a perfect square.
(iii) Prime factorisation of 7688 $=2\times2\times2\times31\times31$
$=2\times(2)^2\times(31)^2$
In order to make the pairs an even number of pairs, we have to multiply 7688 by $2$, then the product will be the perfect square.
Therefore, 2 is the smallest number by which 7688 must be multiplied so that the product is a perfect square.
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