If `A = [k^3 + (k + 2)^3 + (k + 4)^3 + (k + 6)^3 + (k+ 8)^3 + ……(k + 38)^3]` and
`B = [k + (k + 2) + (k + 4) + (k + 6) + (k + 8) + …..(k + 38)]`
Now, if 'A' is divided by 'B', then the remainder will be (for every positive integer k) :

Options

A. A)0
B. B)1
C. C)k
D. D)none of these

Answer

To solve the problem, we need to evaluate the expressions for \( A \) and \( B \) and then find the remainder when \( A \) is divided by \( B \).

### Step 1: Calculate \( A \)

The expression for \( A \) is given as:

\[ A = k^3 + (k + 2)^3 + (k + 4)^3 + (k + 6)^3 + \ldots + (k + 38)^3 \]

This is a sum of cubes of an arithmetic sequence where the first term is \( k \) and the common difference is \( 2 \). The last term is \( k + 38 \).

The number of terms in this sequence can be calculated as follows:

- The first term is \( k \) (when \( n = 0 \)).
- The last term is \( k + 38 \) (when \( n = 19 \)).

Thus, there are \( 20 \) terms in total.

Using the formula for the sum of cubes, we can express \( A \):

\[ A = \sum_{n=0}^{19} (k + 2n)^3 \]

This can be simplified using the formula for the sum of cubes:

\[ \sum_{n=0}^{m-1} n^3 = \left(\frac{m(m-1)}{2}\right)^2 \]

### Step 2: Calculate \( B \)

The expression for \( B \) is:

\[ B = k + (k + 2) + (k + 4) + (k + 6) + \ldots + (k + 38) \]

This is the sum of an arithmetic series where the first term is \( k \), the last term is \( k + 38 \), and the number of terms is \( 20 \).

The sum of an arithmetic series can be calculated using the formula:

\[ S_n = \frac{n}{2} \times (a + l) \]

where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term.

So,

\[ B = \frac{20}{2} \times (k + (k + 38)) = 10 \times (2k + 38) = 20k + 380 \]

### Step 3: Find the remainder of \( A \) divided by \( B \)

Now we need to find \( A \mod B \).

Substituting \( k = 2 \) into \( A \) and \( B \):

1. For \( A \):

\[ A = 2^3 + 4^3 + 6^3 + 8^3 + \ldots + 40^3 \]

Calculating each term:

\[ A = 8 + 64 + 216 + 512 + 1000 + 1728 + 2744 + 4096 + 5832 + 8000 + 10648 + 13824 + 17296 + 21600 + 27000 + 33792 + 40960 + 48600 + 57808 + 68500 \]

Summing these gives:

\[ A = 2 \times 420^2 \]

2. For \( B \):

\[ B = 20 \times 2 + 380 = 400 \]

### Step 4: Calculate \( A \div B \)

Now we can divide \( A \) by \( B \):

\[ \frac{A}{B} = \frac{2 \times 420^2}{400} \]

Calculating this gives:

\[ \frac{A}{B} = 840 \]

Since \( 840 \) is a whole number, the remainder when \( A \) is divided by \( B \) is:

\[ \text{Remainder} = 0 \]

### Final Answer

Thus, the remainder when \( A \) is divided by \( B \) is:

\[ \boxed{0} \]