How many two digit numbers are divisible by 3?

Answer

To find how many two-digit numbers are divisible by 3, we can follow these steps:

### Step 1: Identify the range of two-digit numbers
The smallest two-digit number is 10, and the largest is 99.

### Step 2: Find the first two-digit number divisible by 3
To find the first two-digit number divisible by 3, we start from 10 and check:
- 10 ÷ 3 = 3.33 (not divisible)
- 11 ÷ 3 = 3.67 (not divisible)
- 12 ÷ 3 = 4 (divisible)

So, the first two-digit number divisible by 3 is **12**.

### Step 3: Find the last two-digit number divisible by 3
Now, we find the last two-digit number divisible by 3:
- 99 ÷ 3 = 33 (divisible)

So, the last two-digit number divisible by 3 is **99**.

### Step 4: Form the arithmetic progression (AP)
The two-digit numbers divisible by 3 form an arithmetic progression (AP):
- First term (a) = 12
- Last term (l) = 99
- Common difference (d) = 3

The sequence is: 12, 15, 18, ..., 99.

### Step 5: Use the formula for the nth term of an AP
The nth term of an AP can be calculated using the formula:
\[ l = a + (n - 1) \cdot d \]
Where:
- l = last term
- a = first term
- n = number of terms
- d = common difference

Substituting the known values:
\[ 99 = 12 + (n - 1) \cdot 3 \]

### Step 6: Solve for n
Rearranging the equation:
\[ 99 - 12 = (n - 1) \cdot 3 \]
\[ 87 = (n - 1) \cdot 3 \]

Now, divide both sides by 3:
\[ n - 1 = 29 \]
\[ n = 30 \]

### Conclusion
Thus, there are **30 two-digit numbers that are divisible by 3**.

---