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Find the smallest number by which 1152 must be divided so that it becomes a perfect square. Also, find the square root of the number so obtained.
Given :
The given number is 1152.
To do:
We have to find the smallest number by which 1152 must be divided so that the quotient becomes a perfect square and the number whose square is the resulting number.
Solution:
Perfect Square: A perfect square has each distinct prime factor occurring an even number of times.
$1152=2\times2\times2\times2\times2\times2\times2\times3\times3$
$=(2)^2\times(2)^2\times(2)^2\times2\times(3)^2$
$1152\div2=(2)^2\times(2)^2\times(2)^2\times2\times(3)^2\div2$
$=(2\times2\times2\times3)^2$
$=(24)^2$
In order to make the pairs an even number of pairs, we have to divide 1152 by 2, then the quotient will be the perfect square.
Therefore, 2 is the smallest number by which 1152 must be divided so that the quotient is a perfect square and the number whose square is the resulting number is 24.
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