Question

find the smallest number by which each of the following number should be multiplied so as to get a perfect square also find the square root of the perfect square thus obtained

a)3072 b)4082 c)1452 d)845​

Answer 1

Part 1: 3072

$\begin{array}{r | l}\sf 2 & \sf 3072 \\\cline{2-2} \sf 2 & \sf 1536 \\\cline{2-2} \sf 2 & \sf 768 \\\cline{2-2} \sf 2 & \sf 384 \\\cline{2-2} \sf 2 & \sf 192 \\\cline{2-2} \sf 2 & \sf 96 \\\cline{2-2} \sf 2 & \sf 48 \\\cline{2-2} \sf 2 & \sf 24 \\\cline{2-2} \sf 2 & \sf 12 \\\cline{2-2} \sf 2 & \sf 6 \\\cline{2-2} & \sf 3\end{array}$

$\Longrightarrow \sf 3072 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3$

$\Longrightarrow \sf 3072 = (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times 3$

Here, all other terms except for 3 are paired, therefore, 3072 is not a perfect square. We need to multiply 3 to 3072 to get a perfect square.

⇒ 3072 × 3 = 9216

To find the square root of 9216, let's write down it's prime factorization.

$\begin{array}{r | l}\sf 2 & \sf 9216 \\\cline{2-2} \sf 2 & \sf 4608 \\\cline{2-2} \sf 2 & \sf 2304 \\\cline{2-2} \sf 2 & \sf 1152 \\\cline{2-2} \sf 2 & \sf 576 \\\cline{2-2} \sf 2 & \sf 288 \\\cline{2-2} \sf 2 & \sf 144 \\\cline{2-2} \sf 2 & \sf 72 \\\cline{2-2} \sf 2 & \sf 36 \\\cline{2-2} \sf 2 & \sf 18 \\\cline{2-2} \sf 3 & \sf 9 \\\cline{2-2} & \sf 3\end{array}$

$\sf \Longrightarrow 9216 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3$

Taking square root on both sides we get:

$\sf \Longrightarrow \sqrt{9216} = \sqrt{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3}$

$\sf \Longrightarrow \sqrt{9216} = \sqrt{2^2 \times 2^2 \times 2^2 \times 2^2 \times 2^2 \times 3^3}$

$\sf \Longrightarrow \sqrt{9216} = 2 \times 2 \times 2 \times 2 \times 2 \times 3$

$\sf \Longrightarrow \sqrt{9216} = 4 \times 4 \times 6$

$\sf \Longrightarrow \sqrt{9216} = 96$

Answers for part(a).

⇒ Smallest number to be multiplied to get a perfect square = 3

⇒ Square root of the perfect square = 96.

-----------------------------------

Part 2: 4082

$\begin{array}{r | l}\sf 2 & \sf 4082 \\\cline{2-2}\sf 13 & \sf 2041 \\\cline{2-2}& \sf 157\end{array}$

$\sf \Longrightarrow 4082 = 2 \times 13 \times 157$

2, 13 & 157 are not paired, therefore we have to multiply 4082 by 4082 (2 × 13 × 157 = 4082) to obtain a perfect square.

⇒ 4082 × 4082 = 16662724

And, √16662724 = 4082.

Answers for part(b).

⇒ Smallest number to be multiplied to get a perfect square = 4082

⇒ Square root of the perfect square = 4082

-----------------------------------

Part 3: 1452

$\begin{array}{r | l}\sf 2 & \sf 1452 \\\cline{2-2} \sf 2 & \sf 726 \\\cline{2-2} \sf 3 & \sf 363 \\\cline{2-2} \sf 11 & \sf 121 \\\cline{2-2} & \sf 11 \end{array}$

$\sf \Longrightarrow 1452 = 2 \times 2 \times 3 \times 11 \times 11$

$\sf \Longrightarrow 1452 = (2 \times 2) \times 3 \times (11 \times 11)$

Here, all other terms except for 3 are paired, therefore, we need to multiply 3 to 1452 to get a perfect square.

⇒ 1452 × 3 = 4356

To find the square root of 4356, let's write down its prime factorization.

$\begin{array}{r | l}\sf 2 & \sf 4356 \\\cline{2-2} \sf 2 & \sf 2178 \\\cline{2-2} \sf 3 & \sf 1089 \\\cline{2-2} \sf 3 & \sf 363 \\\cline{2-2} 11 & \sf 121 \\\cline{2-2} & \sf 11 \end{array}$

$\sf \Longrightarrow 4356 = 2 \times 2 \times 3 \times 3 \times 11 \times 11$

Taking the square root on both sides:

$\sf \Longrightarrow \sqrt{4356} = \sqrt{2 \times 2 \times 3 \times 3 \times 11 \times 11}$

$\sf \Longrightarrow \sqrt{4356} = \sqrt{2^2 \times 3^2 \times 11^2}$

$\sf \Longrightarrow \sqrt{4356} = 2 \times 3 \times 11$

$\sf \Longrightarrow \sqrt{4356} = 66$

Answers for part(c).

⇒ Smallest number to be multiplied to get a perfect square = 3

⇒ Square root of the perfect square = 66

-----------------------------------

Part 4: 845

$\begin{array}{r | l}\sf 5 & \sf 845 \\\cline{2-2} \sf 13 & \sf 169 \\\cline{2-2} & \sf 13 \end{array}$

$\sf \Longrightarrow 845 = 5 \times 13 \times 13$

$\sf \Longrightarrow 845 = 5 \times (13 \times 13)$

5 is not paired, therefore we have to multiply 845 by 5 to obtain a perfect square.

⇒ 845 × 5 = 4225

To find the square root of 4225, let's write down its prime factorization.

$\begin{array}{r | l}\sf 5 & \sf 4225 \\\cline{2-2} \sf 5 & \sf 845 \\\cline{2-2} \sf 13 & \sf 169 \\\cline{2-2} & \sf 13 \end{array}$

$\sf \Longrightarrow 4225 = 5 \times 5 \times 13 \times 13$

Taking the square root on both sides:

$\sf \Longrightarrow \sqrt{4225} = \sqrt{5 \times 5 \times 13 \times 13}$

$\sf \Longrightarrow \sqrt{4225} = \sqrt{5^2 \times 13^2}$

$\sf \Longrightarrow \sqrt{4225} = 5 \times 13$

$\sf \Longrightarrow \sqrt{4225} = 65$

Answers for part(d).

⇒ Smallest number to be multiplied to get a perfect square = 5

⇒ Square root of the perfect square = 65

Answer 2

⇒ Square root of the perfect square = 96. 2, 13 & 157 are not paired, therefore we have to multiply 4082 by 4082 (2 × 13 × 157 = 4082) to obtain a perfect square. And, √16662724 = 4082.

hope it helps...