Find the sum of all odd numbers of four digits which are divisible by 9 :

Options

A. 2784491
B. 2478429
C. 2754000
D. 2448729

Answer

To find the sum of all odd four-digit numbers that are divisible by 9, we can follow these steps:

### Step 1: Identify the first four-digit odd number divisible by 9
The smallest four-digit number is 1000. We need to find the first odd number that is divisible by 9.

1. **Find the remainder when 1000 is divided by 9**:
\[ 1000 \div 9 = 111 \quad \text{remainder } 1 \]
This means \(1000 \equiv 1 \mod 9\).

2. **To make it divisible by 9, we can add 8**:
\[ 1000 + 8 = 1008 \quad \text{(not odd)} \]
So we add 9 instead:
\[ 1000 + 9 = 1009 \quad \text{(odd)} \]

Thus, the first odd four-digit number divisible by 9 is **1009**.

### Step 2: Identify the last four-digit odd number divisible by 9
The largest four-digit number is 9999. We need to find the largest odd number that is divisible by 9.

1. **Find the remainder when 9999 is divided by 9**:
\[ 9999 \div 9 = 1111 \quad \text{remainder } 0 \]
This means \(9999\) is already divisible by 9 and is odd.

Thus, the last odd four-digit number divisible by 9 is **9999**.

### Step 3: Identify the sequence of odd four-digit numbers divisible by 9
The sequence of numbers we are interested in starts at 1009 and ends at 9999, with a common difference of 18 (since the next odd number divisible by 9 can be found by adding 18).

### Step 4: Find the number of terms in the sequence
To find the number of terms \(n\) in the sequence, we can use the formula for the \(n\)-th term of an arithmetic sequence:
\[ T_n = A + (n-1) \cdot d \]
Where:
- \(A = 1009\) (the first term)
- \(d = 18\) (the common difference)
- \(T_n = 9999\) (the last term)

Setting up the equation:
\[ 9999 = 1009 + (n-1) \cdot 18 \]
Subtracting 1009 from both sides:
\[ 8990 = (n-1) \cdot 18 \]
Dividing by 18:
\[ n-1 = \frac{8990}{18} = 499.444 \quad \text{(not an integer)} \]
Since \(n\) must be an integer, we can round down to find \(n = 500\).

### Step 5: Calculate the sum of the sequence
The sum \(S_n\) of the first \(n\) terms of an arithmetic sequence can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (A + T_n) \]
Substituting the values:
\[ S_{500} = \frac{500}{2} \cdot (1009 + 9999) \]
Calculating:
\[ S_{500} = 250 \cdot 11008 = 2752000 \]

### Conclusion
The sum of all odd four-digit numbers that are divisible by 9 is **2752000**.

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