Question

find the least number that is to be subtracted from 4000 to make it a perfect square using prime factorization?​

Answer 1

$\green{\sf {Question: }}$

Find the least number that is to be subtracted from 4000 to make it a perfect square using prime factorization?

$\green{\sf {Solution : }}$

Step 1 :

Refer the prime factorization method in the attachment.

Step 2 :

From this we can understand that the square root of the given number (4000) is greater than 63²

Step 3 :

$\red{\sf {63 {}^{2} = 63 \times 63 = 3969 }}$

$\red{\sf{∴4000 - 3969 = 31 }}$

Step 4 :

Now , we need to subtract 31 from 63 in order to find a perfect square .

Hence , 31 is the least number to be subtracted from 4000 to get a perfect square

$\green{\sf {Extra \: Information:}}$

What is a perfect square?

Perfect sqaure is the product of some integer with itself.

Answer 2

For finding the square root we have two methods. The long division method and the prime factorization method.

Here according to the question, we must use prime factorization method.

$\begin{array}{r | l} 2& 4000 \\ \cline{2-2} 2 & 2000 \\ \cline{2-2} 2& 1000 \\ \cline{2-2} 2 & 500 \\ \cline{2-2} 2& 250 \\ \cline{2-2} 5 & 125 \\\cline{2-2} 2& 25 \\ \cline{2-2} & 5 \\ \end{array}$

4000 ⇒ 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5

√4000 ⇒ 2 × 2 × √2 × 5 × √5

√4000 ⇒ 63.24

Now we will find the square of 63.

⇒ 63² = 63 × 63 = 3969

We'll subtract 3969 from 4000 now.

⇒ 4000 - 3969

⇒ 31

∴ 31 is the least number that must be subtracted from 3969 to obtain a perfect square.