Question

Find the least number which must be added to each of the following number so as to get a perfect square.$525$

Answer 1

Hint: We will solve the $525$ by square root method. We will solve $525$ by two methods, in the first method, we will take smaller numbers. In the second method, we will take a bigger number to get the perfect square.


Complete step by step solution:
The given number is $525$
Now, we will solve the $525$ to the square root method

2	$22$
	$525$$ - 4$
$4\underline 2 $	$125$$ - 84$
	$41$
2	$23$
	$525$ $ - 4$
$4\underline 3 $	$125$$ - 129$
	$ - 4$

Remainder$ = 41,$since remainder is not $0$, so $252$ is not a perfect square. So we will add $4$ in the value $252$ to get a perfect square.
The perfect square number is $ = 525 + 4$
$ = 529$
Now, we will verify the value $529$

2	$23$
	$529$ $ - 4$
$4\underline 3 $	$129$$ - 129$
	$0$

Here, remainder is$0$,
So,$529$is a perfect square of $23$.
Additional Information: Properties of square root:
(i) Two square roots can be multiplied$\sqrt 2 $, when multiplied by $\sqrt 3 $, gives $\sqrt 6 $ as a result.
(ii) Two same square roots are multiplied to give anon-square root number. When $\sqrt 3 $ is multiplied by $\sqrt 3 $ we get $3$ as a result.
(iii) If a number ends with an odd number of zeroes, then it cannot have a square root. A square root is only possible for even numbers of zeroes.


Note: In this case, $42 \times 2 = 84$ and $43 \times 3 = 169$. So we choose $3$ as a new digit to be put in divisor and in the quotient. Therefore the remainder here is $0$.