The remainder when `1^3 + 2^3 + 3^3 + …..+ 999^3 + 1000^3` is divided by 13 is :

Options

A. 7
B. 11
C. 12
D. none of these

Answer

To find the remainder when \(1^3 + 2^3 + 3^3 + \ldots + 999^3 + 1000^3\) is divided by 13, we can use the formula for the sum of cubes of the first \(n\) natural numbers. The formula is:

\[ \left( \frac{n(n+1)}{2} \right)^2 \]

For \(n = 1000\), we can substitute into the formula:

1. Calculate \(n(n+1)\):
\[ n(n+1) = 1000 \times 1001 = 1001000 \]

2. Divide by 2:
\[ \frac{1001000}{2} = 500500 \]

3. Square the result:
\[ (500500)^2 \]

Now we need to find the remainder of \((500500)^2\) when divided by 13. Instead of calculating \((500500)^2\) directly, we can first find \(500500 \mod 13\).

4. Calculate \(500500 \mod 13\):
- First, find \(1000 \mod 13\):
\[ 1000 \div 13 \approx 76.923 \quad \Rightarrow \quad 76 \times 13 = 988 \quad \Rightarrow \quad 1000 - 988 = 12 \]
So, \(1000 \equiv 12 \mod 13\).
- Next, find \(1001 \mod 13\):
\[ 1001 \div 13 \approx 77.0 \quad \Rightarrow \quad 77 \times 13 = 1001 \quad \Rightarrow \quad 1001 - 1001 = 0 \]
So, \(1001 \equiv 0 \mod 13\).

5. Now calculate \(500500 \mod 13\):
\[ 500500 \equiv \frac{1000 \times 1001}{2} \equiv \frac{12 \times 0}{2} \equiv 0 \mod 13 \]

6. Finally, calculate \((500500)^2 \mod 13\):
\[ (500500)^2 \equiv 0^2 \equiv 0 \mod 13 \]

Thus, the remainder when \(1^3 + 2^3 + 3^3 + \ldots + 999^3 + 1000^3\) is divided by 13 is \(0\).

### Final Answer:
The remainder is \(0\).