What is the least natural number that should be added to the product of 30 and 31 so that the sum obtained is a perfect square?

Options

A. 10
B. 3
C. 30
D. 31

Answer

To solve the problem, we need to find the least natural number that should be added to the product of 30 and 31 so that the resulting sum is a perfect square. Let's break this down step by step.

### Step 1: Calculate the product of 30 and 31
First, we need to calculate the product of 30 and 31.

\[ 30 \times 31 = 930 \]

### Step 2: Identify the nearest perfect square greater than 930
Next, we need to find the nearest perfect square that is greater than 930. We can start by calculating the square root of 930.

\[ \sqrt{930} \approx 30.5 \]

The next whole number after 30.5 is 31. So, we calculate \(31^2\):

\[ 31^2 = 961 \]

### Step 3: Determine the least natural number to be added
Now, we need to find the least natural number \(K\) that should be added to 930 to make it equal to 961.

\[ K = 961 - 930 \]

Calculating this gives:

\[ K = 31 \]

### Conclusion
Thus, the least natural number that should be added to the product of 30 and 31 to make the sum a perfect square is **31**.

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