How many two -digit numbers are divisible by 3?

Options

A. 25
B. 30
C. 32
D. 36

Explanation

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 two-digit number is 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 and round up to the nearest whole number.

\[ 10 \div 3 = 3.33 \quad \text{(round up to 4)} \]
Now, multiply 4 by 3 to find the smallest two-digit number divisible by 3:
\[ 4 \times 3 = 12 \]
So, the smallest two-digit number divisible by 3 is 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.

\[ 99 \div 3 = 33 \]
Now, multiply 33 by 3 to find the largest two-digit number divisible by 3:
\[ 33 \times 3 = 99 \]
So, the largest two-digit number divisible by 3 is 99.

### Step 4: Form an arithmetic progression (AP)
The two-digit numbers divisible by 3 form an arithmetic progression where:
- The first term \(a = 12\)
- The last term \(l = 99\)
- The common difference \(d = 3\)

### Step 5: Find the number of terms in the AP
The formula for the nth term of an arithmetic progression is given by:
\[ l = a + (n-1) \cdot d \]
We can rearrange this to solve for \(n\):
\[ n = \frac{l - a}{d} + 1 \]
Substituting the known values:
\[ n = \frac{99 - 12}{3} + 1 \]
Calculating this gives:
\[ n = \frac{87}{3} + 1 = 29 + 1 = 30 \]

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

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