Question

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

1. 402
2. 1989
3. 3250
4. 825
5. 4000

Answer 1

(i) The square root of $402$ can be calculated by long division method as follows.

| 20
---|---
2| $\bar4\bar0\bar2$
$-4$
4| 02
00
| 2

The remainder is $2$.
It represents that the square of $20$ is less than $402$ by $2$.
Therefore, a perfect square will be obtained by subtracting $2$ from the given number $402$.
Therefore, required perfect square = $402 - 2 = 400$
And, $\sqrt{400} = 20$

---

(ii) The square root of $1989$ can be calculated by long division method as follows.

| 44
---|---
4| $\bar1\bar9\bar8\bar9$
$-16$
84| 389
336
| 53

The remainder is $53$.
It represents that the square of $44$ is less than $1989$ by $53$.
Therefore, a perfect square will be obtained by subtracting $53$ from the given number $1989$.
Therefore, required perfect square = $1989 - 53 = 1936$
And, $\sqrt{1936} = 44$

---

(iii) The square root of $3250$ can be calculated by long division method as follows.

| 57
---|---
5| $\bar3\bar2\bar5\bar0$
$-25$
107| 750
749
| 1

The remainder is $1$.
It represents that the square of $57$ is less than $3250$ by $1$.
Therefore, a perfect square can be obtained by subtracting $1$ from the given number $3250$.
Therefore, required perfect square = $3250 - 1 = 3249$
And, $\sqrt{3249} = 57$

---

(iv) The square root of $825$ can be calculated by long division method as follows.

| 28
---|---
2| $\bar8\bar2\bar5$
-4
48| 425
384
| 41

The remainder is $41$.
It represents that the square of $28$ is less than $825$ by $41$.
Therefore, a perfect square can be calculated by subtracting.

---

(v) $4000$ We know that, if we subtract the remainder from the number, we get a perfect square.

| 63
---|---
6| $\bar4\bar0\bar0\bar0$
$-36$
123| 400
-369
| 31

Here, we get remainder $31$.
Therefore $31$ must be subtracted from $4000$ to get a perfect square.
$\therefore$ $4000 – 31 = 3969$