Question

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

1. 525
2. 1750
3. 252
4. 1825
5. 6412

Answer 1

(i) The square root of $525$ can be calculated by long division method as follows.

| 22
---|---
2| $\bar 5\bar 2 \bar 5$
$-4$
42| 125
84
| 41

The remainder is $41$.
It represents that the square of $22$ is less than $525$.
Next number is $23$ and $23^2$= $529$
Hence, number to be added to $525 = 232- 525 = 529 - 525 = 4$
The required perfect square is $529$ and $\sqrt{529} = 23$

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(ii) The square root of $1750$ can be calculated by long division method as follows.

| 41
---|---
4| $\bar 1\bar 7 \bar 5\bar0$
$-4$
81| 150
81
| 69

The remainder is $69$.
It represents that the square of $41$ is less than $1750$.
The next number is $42$ and $42^2$ = $1764$
Hence, number to be added to $1750$ = 422 $- 1750 = 1764 - 1750 = 14$
The required perfect square is $1764$ and $\sqrt{1764} = 42$

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(iii) The square root of $252$ can be calculated by long division method as follows.

| 15
---|---
1| $\bar 2\bar 5 \bar 2$
$-1$
25| 152
125
| 27

The remainder is $27$.
It represents that the square of $15$ is less than $252$.
The next number is $16$ and $162$
= $256$
Hence, number to be added to $252$ = 162$$- 252 = 256 - 252 = 4
The required perfect square is $256$ and $\sqrt{256}$ = $16$

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(iv) The square root of $1825$ can be calculated by long division method as follows.

| 42
---|---
4| $\bar 1\bar 8 \bar 2\bar5$
$-16$
82| 225
164
| 61

The remainder is $61$.
It represents that the square of $42$