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

Options

A. 0
B. 3
C. 8
D. 15

Answer

To find the smallest number \( X \) such that when added to the sum of the squares of 9 and 10, it results in a perfect square, we can follow these steps:

### Step 1: Calculate the sum of the squares of 9 and 10
First, we need to find the squares of 9 and 10.

\[ 9^2 = 81 \]
\[ 10^2 = 100 \]

Now, we add these two results together:

\[ 81 + 100 = 181 \]

### Step 2: Identify the next perfect square greater than 181
Next, we need to find the smallest perfect square that is greater than 181. The perfect squares around 181 are:

\[ 13^2 = 169 \quad (\text{less than } 181) \]
\[ 14^2 = 196 \quad (\text{greater than } 181) \]

So, the next perfect square after 181 is \( 196 \).

### Step 3: Calculate the smallest number \( X \)
Now, we need to find \( X \) such that:

\[ 181 + X = 196 \]

To find \( X \), we rearrange the equation:

\[ X = 196 - 181 \]
\[ X = 15 \]

### Conclusion
Thus, the smallest number \( X \) that needs to be added to the sum of the squares of 9 and 10 to make it a perfect square is:

\[ \boxed{15} \]

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