Question

Find the least number which must be added from each of the following so as to get a perfect square. Also find the square root of the perfect square so obtained.
(i)525 (ii) 1750 (iii) 252 (iv) 1825 (v) 6412

Answer 1

Hint: Find the greatest perfect square for a number and subtract from it.

The approach to find the least number that should be added to the given numbers in order to make them perfect square is that we need to find a greater perfect number which is closest to the given number.
(i) 525
The closest perfect square number to 525 is 576 hence difference between two of them is $576 - 525 = 51$
Thus 51 must be added to 525 make it a perfect square thus $\sqrt {576} = 24$

(ii) 1750
The closest perfect square number to 1750 is 1764 hence difference between two of them is
$1764 - 1750 = 14$
Thus 14 must be added to 1750 make it a perfect square thus $\sqrt {1764} = 42$

(iii) 252
The closest perfect square number to 252 is 256 hence difference between two of them is $256 - 252 = 4$
Thus 4 must be added to 252 make it a perfect square thus $\sqrt {256} = 16$

(iv) 1825
The closest perfect square number to 1825 is 1849 hence difference between two of them is $1849 - 1825 = 24$
Thus 24 must be added to 1825 make it a perfect square thus $\sqrt {1849} = 43$

(v) 6412
The closest perfect square number to 6412 is 6561 hence difference between two of them is $6561 - 6412 = 149$
Thus 149 must be added to 6412 make it a perfect square thus $\sqrt {6561} = 81$

Note: Whenever we need to solve such problem statements we simply need to find the closest perfect square to that given number thus eventually their difference will give us the number that should be added in order to make the perfect square.