How many two digit numbers are divisible by 4?

Answer

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

### Step 1: Identify the range of two-digit numbers
The two-digit numbers range from 10 to 99.

### Step 2: Find the smallest two-digit number divisible by 4
To find the smallest two-digit number divisible by 4, we can start from 10 and check:
- 10 ÷ 4 = 2.5 (not divisible)
- 11 ÷ 4 = 2.75 (not divisible)
- 12 ÷ 4 = 3 (divisible)

So, the smallest two-digit number divisible by 4 is **12**.

### Step 3: Find the largest two-digit number divisible by 4
Now, we need to find the largest two-digit number divisible by 4. We can start from 99 and check:
- 99 ÷ 4 = 24.75 (not divisible)
- 98 ÷ 4 = 24.5 (not divisible)
- 97 ÷ 4 = 24.25 (not divisible)
- 96 ÷ 4 = 24 (divisible)

So, the largest two-digit number divisible by 4 is **96**.

### Step 4: List the two-digit numbers divisible by 4
Now, we can list the two-digit numbers divisible by 4 starting from 12 to 96. These numbers form an arithmetic progression (AP) where:
- First term (a) = 12
- Common difference (d) = 4
- Last term (l) = 96

### Step 5: Use the formula for the nth term of an AP
The nth term of an AP can be calculated using the formula:
\[ T_n = a + (n - 1) \times d \]

Setting \( T_n = 96 \):
\[ 96 = 12 + (n - 1) \times 4 \]

### Step 6: Solve for n
Rearranging the equation:
\[ 96 - 12 = (n - 1) \times 4 \]
\[ 84 = (n - 1) \times 4 \]
\[ n - 1 = \frac{84}{4} \]
\[ n - 1 = 21 \]
\[ n = 21 + 1 \]
\[ n = 22 \]

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

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