How many two-digit numbers are divisible by 3?

Options

A. 30
B. 29
C. 85
D. 99

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 two-digit numbers range from 10 to 99.

### Step 2: Find the smallest two-digit number divisible by 3
To find the smallest two-digit number divisible by 3, we can divide 10 by 3:
- \(10 \div 3 = 3\) remainder \(1\)
Thus, the smallest two-digit number divisible by 3 is \(10 + (3 - 1) = 12\).

### Step 3: Find the largest two-digit number divisible by 3
To find the largest two-digit number divisible by 3, we divide 99 by 3:
- \(99 \div 3 = 33\) remainder \(0\)
Thus, the largest two-digit number divisible by 3 is 99.

### Step 4: List the two-digit numbers divisible by 3
The two-digit numbers divisible by 3 form an arithmetic sequence starting from 12 to 99 with a common difference of 3. The sequence is:
12, 15, 18, ..., 99.

### Step 5: Determine the number of terms in the sequence
To find the number of terms in this arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:
\[ a_n = a + (n-1)d \]
Where:
- \(a\) is the first term (12)
- \(d\) is the common difference (3)
- \(a_n\) is the last term (99)

Setting up the equation:
\[ 99 = 12 + (n-1) \cdot 3 \]

### Step 6: Solve for \(n\)
Rearranging the equation:
\[ 99 - 12 = (n-1) \cdot 3 \]
\[ 87 = (n-1) \cdot 3 \]
\[ n-1 = 29 \]
\[ n = 30 \]

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

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