For each positive integer k , let `K_(0)` be the sum of all positive odd integers less than K. For example , `6_(@) =5+3+1=9` and `7_(@)=5+3+1=9.` what is the value of `17 _(@) xx 4 _(@)`?

Options

A. 16
B. 144
C. 256
D. 324

Answer

To solve the problem, we need to find the value of \( 17_(@) \times 4_(@) \) where \( k_0 \) is defined as the sum of all positive odd integers less than \( k \).

### Step 1: Calculate \( 17_(@) \)

1. Identify all positive odd integers less than 17:
- The odd integers less than 17 are: 15, 13, 11, 9, 7, 5, 3, and 1.

2. Sum these integers:
\[ 15 + 13 + 11 + 9 + 7 + 5 + 3 + 1 \]

3. Calculate the sum step by step:
- \( 15 + 13 = 28 \)
- \( 28 + 11 = 39 \)
- \( 39 + 9 = 48 \)
- \( 48 + 7 = 55 \)
- \( 55 + 5 = 60 \)
- \( 60 + 3 = 63 \)
- \( 63 + 1 = 64 \)

Thus, \( 17_(@) = 64 \).

### Step 2: Calculate \( 4_(@) \)

1. Identify all positive odd integers less than 4:
- The odd integers less than 4 are: 3 and 1.

2. Sum these integers:
\[ 3 + 1 \]

3. Calculate the sum:
- \( 3 + 1 = 4 \)

Thus, \( 4_(@) = 4 \).

### Step 3: Multiply \( 17_(@) \) and \( 4_(@) \)

Now, we multiply the results from the previous steps:
\[ 17_(@) \times 4_(@) = 64 \times 4 \]

4. Calculate the product:
- \( 64 \times 4 = 256 \)

Thus, the final result is:
\[ \boxed{256} \]