Find the smallest number such that if it is added to sum of squares of 9 and 10, then complete square is obtained:

Question

Accepted Answer

To solve the problem, we need to find the smallest number that, when added to the sum of the squares of 9 and 10, results in a perfect square.

First, calculate the sum of the squares of 9 and 10:

$$ 9^2 = 81 $$
$$ 10^2 = 100 $$

Adding these together gives:

$$ 81 + 100 = 181 $$

We need to find the smallest number $ x $ such that $ 181 + x $ is a perfect square. Let's denote the perfect square as $ n^2 $.

We start by checking the perfect squares around 181:

- $ 13^2 = 169 $ (too small)
- $ 14^2 = 196 $

Since 196 is the next perfect square greater than 181, we calculate:

$$ 196 - 181 = 15 $$

Thus, adding 15 to 181 results in 196, which is a perfect square ($ 14^2 $).

Therefore, the smallest number that needs to be added to 181 to make it a perfect square is 15. This confirms that Option 4 is correct.

Now, let's explain why the other options are incorrect:

1. **Option 1 (0):** Adding 0 to 181 gives 181, which is not a perfect square.
2. **Option 2 (3):** Adding 3 to 181 gives 184, which is not a perfect square.
3. **Option 3 (8):** Adding 8 to 181 gives 189, which is not a perfect square.

Only Option 4 (15) results in a perfect square (196), confirming it as the correct answer.