Question

The number $\left( {{{10}^n} - 1} \right)$ is divisible by 11 for ____
A) $n \in N$
B) Odd values of n
C) Even values of n
D) n is the multiple of 11

Answer 1

Hint: When we substitute any value in the place of n in $\left( {{{10}^n} - 1} \right)$ the result will have only 9’s. The no. of 9’s in the result is determined by the value of n. We just have to find for a number to be divisible by 11 how many no. of 9’s it should have.


Complete step-by-step Solution:
We are given that $\left( {{{10}^n} - 1} \right)$ is divisible by 11. We have to find for what value of n, $\left( {{{10}^n} - 1} \right)$ is divisible by 11.
When
 $
  n = 1, \\
  \left( {{{10}^n} - 1} \right) = \left( {{{10}^1} - 1} \right) \\
   = 9 \\
 $
9 is not divisible by 11.
When
 $
  n = 2, \\
  \left( {{{10}^n} - 1} \right) = \left( {{{10}^2} - 1} \right) \\
   = 99 \\
 $
99 is divisible by 11.
When
 $
  n = 3, \\
  \left( {{{10}^n} - 1} \right) = \left( {{{10}^3} - 1} \right) \\
   = 999 \\
 $
999 is not divisible by 11.
When
 $
  n = 4, \\
  \left( {{{10}^n} - 1} \right) = \left( {{{10}^4} - 1} \right) \\
   = 9999 \\
 $
9999 is divisible by 11.
Therefore, when the number of 9’s in $\left( {{{10}^n} - 1} \right)$ is even, then the number is divisible by 11.
So, to get an even number of 9’s in the result, the value of ‘n’ must be even.
Therefore, from among the options given in the question option C is correct, which is the number $\left( {{{10}^n} - 1} \right)$ is divisible by 11 for even values of n. For even values of n, the number $\left( {{{10}^n} - 1} \right)$ consists of even numbers of nines and hence it will be divisible by 11.


Note: Division is splitting into equal parts or groups and is opposite of multiplication. For a number y to be divisible by another number x, y must be a multiple of x and the remainder must be zero.