How many natural numbers lie between `18^2 and 19^2` ?

Options

A. 30
B. 37
C. 35
D. 36

Answer

To find how many natural numbers lie between \(18^2\) and \(19^2\), we can follow these steps:

### Step 1: Calculate \(18^2\) and \(19^2\)
First, we need to calculate the squares of 18 and 19.

\[ 18^2 = 18 \times 18 = 324 \]
\[ 19^2 = 19 \times 19 = 361 \]

### Step 2: Identify the range of natural numbers
Next, we need to identify the natural numbers that lie between \(324\) and \(361\). The natural numbers between these two squares do not include \(324\) and \(361\) themselves.

The smallest natural number greater than \(324\) is \(325\), and the largest natural number less than \(361\) is \(360\).

### Step 3: Count the natural numbers in the range
Now, we will count the natural numbers from \(325\) to \(360\).

To find the total count, we can use the formula:

\[ \text{Count} = \text{Last number} - \text{First number} + 1 \]

Substituting the values:

\[ \text{Count} = 360 - 325 + 1 = 36 \]

### Conclusion
Thus, the total number of natural numbers that lie between \(18^2\) and \(19^2\) is \(36\).

---