Find the smallest number by which the following number must be multiplied so as to get a perfect square? Also find the number whose square is the new number.
7776

Given :

The given number is 7776.

To do :

We have to find the number to be multiplied by 7776 to make it a perfect square.

Solution :

The prime factorization of 7776 is as follows

Prime factors of 7776 $= 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 $

$7776 = 2^5 \times 3^5$

$7776 = (2^2 \times 3^2)^2 \times 2 \times 3$

As we can see, if we multiply the number $7776$ by $2 \times 3$, then the number will become a perfect square.

So for 7776 to be a perfect square, the smallest number to be multiplied is $2\times3=6$.

$7776\times6 = (2^2 \times 3^2)^2 \times 6 \times 6$

$46656 = (4\times9\times6)^2$

 $46656 = (216)^2$.

Therefore, the smallest number it needs to be multiplied to make it a perfect square is 6.

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Updated on: 10-Oct-2022

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