How many two digit number are divisible by 9?

Options

A. 9
B. 8
C. 10
D. 1

Answer

To find how many two-digit numbers are divisible by 9, we can follow these steps:

### Step 1: Identify the smallest two-digit number divisible by 9.
The smallest two-digit number is 10. The first two-digit number that is divisible by 9 is 18.

### Step 2: Identify the largest two-digit number divisible by 9.
The largest two-digit number is 99. Since 99 is divisible by 9, it is our largest two-digit number in this range.

### Step 3: List the two-digit numbers divisible by 9.
The two-digit numbers divisible by 9 are: 18, 27, 36, 45, 54, 63, 72, 81, 90, and 99.

### Step 4: Determine the sequence of these numbers.
These numbers form an arithmetic progression (AP) where:
- The first term (A) is 18.
- The last term (AN) is 99.
- The common difference (D) is 9.

### Step 5: 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:
\[ 99 = 18 + (N - 1) \cdot 9 \]

### Step 6: Solve for N.
Rearranging the equation:
\[ 99 - 18 = (N - 1) \cdot 9 \]
\[ 81 = (N - 1) \cdot 9 \]
Now, divide both sides by 9:
\[ N - 1 = \frac{81}{9} \]
\[ N - 1 = 9 \]
Adding 1 to both sides gives:
\[ N = 10 \]

### Conclusion:
Thus, there are **10 two-digit numbers** that are divisible by 9.

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