Chapter 11.7, Problem 6CYU

To determine

To find a counterexample to disprove that $3^n+1$ is divisible by 4.

Answer to Problem 6CYU

For $n=2$ .

Explanation of Solution

Given:

  $3^n+1$ is divisible by 4.

Calculation:

In order to find a counterexample to disprove that $3^n+1$ is divisible by 4.

First, let’s check this is true for $n=1$ .

For $n=1$

  $\begin{array}{c}{3}^1+1=3+1\\ =4\end{array}$

And 4 is divisible by 4. Thus, the given identity is true for $n=1$ .

Thus, formula is true for $n=1$ .

But, for $n=2$ :

  $\begin{array}{c}{3}^2+1=9+1\\ =10\end{array}$

But 10 is not divisible by 4. Thus, the given identity is not true for $n=2$ .

Thus, counter example to disprove the given statement is for $n=2$ .