Question

${11}^3+{12}^3+...{20}^3$

• A. is divisible by $5$
• B. is an odd integer divisible by $5$
• C. is an even integer which is not divisible by $5$
• D. is an odd integer which is not divisible by $5$

Answer 1

The correct option is B

is an odd integer divisible by $5$

Explanation for the correct option:

Step 1. Add and subtract $\left(1^3+2^3+3^3+4^3+.....+9^3\right)$ in the given series ${11}^3+{12}^3+...{20}^3$

$=\left(1^3+2^3+3^3+4^3+.....+9^3+{10}^3+{11}^3+......+{20}^3\right)-\left(1^3+2^3+3^3+4^3+.....+9^3\right)$ ……….(1)

Here, $n=20$ and $9$

Step 2. Find the solution of equation (1) by applying sum formula of $n$ natural numbers:

${11}^3+{12}^3+...{20}^3={\left[\frac{20\left(20+1\right)}{2}-\frac{9\left(9+1\right)}{2}\right]}^2$ $\left[\mathbf{\because }\left(1^3+2^3+3^3+4^3+.....+n^3\right)\mathbf{=}{\left[\frac{n\left(n+1\right)}{2}\right]}^{\mathbf{2}}\right]$

$=\left[\frac{20\times 21}{2}-\frac{9\times 10}{2}\right]$

$={\left(210\right)}^2-{\left(45\right)}^2$

$=44100-2025$

$=42075$

$\therefore 42075$ is an odd integer divisible by 5.

Hence, Option ‘B’ is Correct.