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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.