How many two digit numbers are divisible by 4?

Options

A. A. 21
B. B. 22
C. C. 24
D. D. 25

Answer

To find how many two-digit numbers are divisible by 4, we will follow these steps:

### Step 1: Identify the smallest and largest two-digit numbers divisible by 4.
The smallest two-digit number is 10, and the largest two-digit number is 99. We need to find the first two-digit number greater than or equal to 10 that is divisible by 4.

- The smallest two-digit number divisible by 4 is 12 (since 12 ÷ 4 = 3).
- The largest two-digit number divisible by 4 is 96 (since 96 ÷ 4 = 24).

### Step 2: List the two-digit numbers divisible by 4.
The two-digit numbers divisible by 4 form an arithmetic progression (AP) where:
- First term (A) = 12
- Common difference (D) = 4
- Last term (An) = 96

The sequence of two-digit numbers divisible by 4 is: 12, 16, 20, 24, ..., 96.

### Step 3: Use the formula for the nth term of an arithmetic progression.
The formula for the nth term of an AP is given by:
\[ An = A + (n - 1) \cdot D \]
Substituting the known values:
\[ 96 = 12 + (n - 1) \cdot 4 \]

### Step 4: Solve for n.
Rearranging the equation:
\[ 96 - 12 = (n - 1) \cdot 4 \]
\[ 84 = (n - 1) \cdot 4 \]
Dividing both sides by 4:
\[ 21 = n - 1 \]
Adding 1 to both sides:
\[ n = 22 \]

### Conclusion
Thus, there are **22 two-digit numbers** that are divisible by 4.