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For the following number find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also find the square root of the square number so obtained (i) 2028

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Last updated date: 20th Jun 2024
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Answer
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Hint:To solve such questions we need to prime factorize the number. Once the prime factors are generated we have to make a pair since it is a square root and find the one whose number doesn’t have a pair.

Complete step-by-step answer:
We are given that 2028 is a number and we have to find the smallest whole number which is to be multiplied to make it a perfect square.
First step to solve such questions is the prime factorization. Once we get the factors of the number we make pairs.
         $$
           \begin{array}{|l}
           \llap{2~~~~} 2028 \\ \hline
           \llap{2~~~~} 1014 \\ \hline
           \llap{3~~~~} 507 \\ \hline
           \llap{13~~~~} 169 \\ \hline
           \llap{13~~~~} 13 \\ \hline
                        1
          \end{array}
        $$
Therefore, by prime factorization we get 2028=2$ \times $2$ \times $3$ \times $13$ \times $13
Since, we have a pair of 2’s and a pair of 13’s but not a pair of 3. So we have to multiply this number by 3 to make it a perfect square.
Since, 3 does not come in a pair we multiply by 3 to make it a pair as shown,
$2028 \times 3 = 2 \times 2 \times 3 \times 13 \times 13$
Forming pairs we get,
$6048 = 2 \times 2 \times 3 \times 3 \times 13 \times13$
$6048 =2^2 \times 3^2 \times 13^2$
Taking square root of 6048 we get,
$6048 = \sqrt{ 2^2 \times 3^2 \times 13^2}$
The smallest whole number to be multiplied is 3 as shown above.
And the square of the new number is 78.$ \times $

Note:While taking square roots we have to make pairs in case we have to calculate the cube root of a number we cannot take pairs but instead we take triplets of factors.