Chapter 10, Problem 21PT

To determine

Prove that ${11}^n-1$ is divisible by 10 for all positive integers.

Answer to Problem 21PT

The statement ${11}^n-1$ is divisible by 10 for all positive integers is a true statement.

Explanation of Solution

Given:

Prove that ${11}^n-1$ is divisible by 10 for all positive integers.

Concept Used:

Substitute n = 1, 2, 3, 4 ……n and check each cases the number is divisible by 10 or not .

Calculation:

  $\begin{array}{l}n=1\Rightarrow {11}^1-1=10\Rightarrow 10\text{is}\text{divisible}\text{by}10\\ n=2\Rightarrow {11}^2-1=120\Rightarrow 120\text{is}\text{divisible}\text{by}10\\ n=3\Rightarrow {11}^3-1=1330\Rightarrow 1330\text{is}\text{divisible}\text{by}10\\ n=4\Rightarrow {11}^4-1=14640\Rightarrow 14640\text{is}\text{divisible}\text{by}10\end{array}$

Now we can write that $n=n\Rightarrow {11}^n-1\Rightarrow \text{is}\text{always}\text{divisible}\text{by}10$

Thus, the statement ${11}^n-1$ is divisible by 10 for all positive integers is a true statement.