Question

Find the least number which must be added to 6412 so as to get a perfect square. Also find the square root of the perfect square.

A) 147; 81

B) 147; 83

C) 149; 81

D) 149; 83

Answer 1

Hint: Try to find the perfect squares which are the nearest to given number and compare the squares with the given number if it is less than the given number then take the next perfect square because here, we have to add something in the given number in order to make it perfect square of any number. Then we will find the square root.

Complete step by step answer:

We are given a number $6412$. 

We have to find the least number which must be added to $6412$ so as to get a perfect square and also square root of the perfect square.

Now, we think about the nearest perfect square number and compare with the given number.

The first two digits are $64$ and it is a square of the number $8$. It means the number at the tenth place of square root will be $8$.

Now, we think about the unit place of the square root number.

Let the number at the unit place is $0$.

If we square off the number $80$, the answer will be $6400$.

But it is less than the given number so this cannot be our number.

Let the number at the unit's place is $1$.

If we square off the number $81$, the answer will be $6561$.

This number is greater than the given number. It means we have to subtract the given number from $6561$ to get that value which can be added to the number $6412$ to make it a perfect square.

Therefore, $6561 - 6412 = 149$

Hence, $149$ is the least number which must be added to $6412$ so as to get a perfect square.

The required number is $6412 + 149 = 6561$.

Now we evaluate the square root of the number $6561$.

On factorisation, we get

$\sqrt {6561}  = \sqrt {3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3} $

Evaluate the square root. The numbers which are in the pair will come out from the square root.

That is,

 \[  \sqrt {6561}  = 3 \times 3 \times 3 \times 3\]

 \[ \Rightarrow \sqrt {6561}  = 81  \]

Hence, option (C) is correct.

Note:

We can directly check from the options. If we add the first number given in the options to the given number and it will give the perfect square then that option will be our answer.

Option-(A) 

$6412 + 147 = 6559$, But $6559$ is not a perfect square.

Option-(B) 

$6412 + 147 = 6559$, But $6559$ is not a perfect square.

Option-(C) 

$6412 + 149 = 6561$, But $6561$ is not a perfect square of $81$.

Hence, option (C) is correct.