Find the smallest number by which each of the following numbers should be multiplied so as to get a perfect square. Also, find the square root of the perfect square thus obtained. a) $3072$ b) $4802$ c) $1452$ d) $845$

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Q. Find the smallest number by which each of the following numbers should be multiplied so as to get a perfect square. Also, find the square root of the perfect square thus obtained. a)

3072

4802

1452

845

Factorize $3072$ : To find the smallest number by which $3072$ should be multiplied to get a perfect square, we first factorize $3072$ into its prime factors. $\newline$ $3072 = 2^{10} \times 3$
Make $3072$ a perfect square: To make $3072$ a perfect square, we need to have pairs of prime factors. We have an even power of $2$ , which is good, but we need another $3$ to pair up with the existing one. $\newline$ So, we need to multiply $3072$ by $3$ to get a perfect square. $\newline$ $3072 \times 3 = 2^{10} \times 3^2$
Square root of $3072$ : The square root of the perfect square obtained by multiplying $3072$ by $3$ is: $\sqrt{(2^{10} \times 3^{2})} = 2^{5} \times 3 = 32 \times 3 = 96$
Factorize $4802$ : Now, we move on to the number $4802$ . We factorize $4802$ into its prime factors. $\newline$ $4802 = 2 \times 2401 = 2 \times 7^4$
Make $4802$ a perfect square: To make $4802$ a perfect square, we need to multiply it by $2$ to pair up with the existing $2$ . $\newline$ $4802 \times 2 = 2^2 \times 7^4$
Square root of $4802$ : The square root of the perfect square obtained by multiplying $4802$ by $2$ is: $\newline$ $\sqrt{2^2 \times 7^4} = 2 \times 7^2 = 2 \times 49 = 98$
Factorize $1452$ : Next, we consider the number $1452$ . We factorize $1452$ into its prime factors. $\newline$ $1452 = 2^2 \times 3 \times 11^2$
Make $1452$ a perfect square: To make $1452$ a perfect square, we need to multiply it by $3$ to pair up with the existing $3$ . $\newline$ $1452 \times 3 = 2^2 \times 3^2 \times 11^2$
Square root of $1452$ : The square root of the perfect square obtained by multiplying $1452$ by $3$ is: $\newline$ $\sqrt{(2^2 \times 3^2 \times 11^2)} = 2 \times 3 \times 11 = 6 \times 11 = 66$
Factorize $845$ : Finally, we look at the number $845$ . We factorize $845$ into its prime factors. $845 = 5 \times 13^2$
Make $845$ a perfect square: To make $845$ a perfect square, we need to multiply it by $5$ to pair up with the existing $5$ . $\newline$ $845 \times 5 = 5^2 \times 13^2$
Square root of $845$ : The square root of the perfect square obtained by multiplying $845$ by $5$ is: $\newline$ $\sqrt{(5^2 \times 13^2)} = 5 \times 13 = 65$