The number of perfect square numbers between 50 and 1000 is

Options

A. 21
B. 22
C. 23
D. 24

Answer

To find the number of perfect square numbers between 50 and 1000, we can follow these steps:

### Step 1: Identify the smallest perfect square greater than 50.
The smallest integer whose square is greater than 50 is 8, since:
\[ 8^2 = 64 \]
So, the first perfect square number we consider is 64.

### Step 2: Identify the largest perfect square less than or equal to 1000.
The largest integer whose square is less than or equal to 1000 is 31, since:
\[ 31^2 = 961 \]
So, the largest perfect square number we consider is 961.

### Step 3: List the perfect squares from 8 to 31.
The perfect squares from 8 to 31 are:
- \( 8^2 = 64 \)
- \( 9^2 = 81 \)
- \( 10^2 = 100 \)
- \( 11^2 = 121 \)
- \( 12^2 = 144 \)
- \( 13^2 = 169 \)
- \( 14^2 = 196 \)
- \( 15^2 = 225 \)
- \( 16^2 = 256 \)
- \( 17^2 = 289 \)
- \( 18^2 = 324 \)
- \( 19^2 = 361 \)
- \( 20^2 = 400 \)
- \( 21^2 = 441 \)
- \( 22^2 = 484 \)
- \( 23^2 = 529 \)
- \( 24^2 = 576 \)
- \( 25^2 = 625 \)
- \( 26^2 = 676 \)
- \( 27^2 = 729 \)
- \( 28^2 = 784 \)
- \( 29^2 = 841 \)
- \( 30^2 = 900 \)
- \( 31^2 = 961 \)

### Step 4: Count the number of perfect squares.
Now, we count the perfect squares from \( 8^2 \) to \( 31^2 \):
- The integers from 8 to 31 are: 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31.

To find the count:
- The total count of integers from 8 to 31 is:
\[ 31 - 8 + 1 = 24 \]

### Conclusion:
Thus, the number of perfect square numbers between 50 and 1000 is **24**.

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