Question

For every +ve integer 'n' prove that $7^n-3^n$ is divisible by $4$.

Answer 1

$P\left(\eta \right):7^{\eta }-3^{\eta }$ is divisible by $4$

Step I : $\eta -1$

$\Rightarrow 7^{\eta }-3^{\eta }=7^{\prime }-3^{\prime }=4$

$\Rightarrow P\left(1\right)$ is true

Step II : Assume $P\left(\eta \right)$ is true for $\eta =m$

$\Rightarrow 7^m-3^m$ is divisible by $4$.

$\Rightarrow 7^m=3^m+4k;K\in N$

Step III : $7^{m+1}-3^{m+1}$

$={7.7}^m-3^{m+1}$

$=7.(3^m+4k)-3^{m+1}$

$={7.3}^m-3^{m+1}+28k$

$=3^m\left(4\right)+28k$

$=4[3^k+7k]$

$=4k^{\prime }$

$\Rightarrow P(m+1)$ is true

I, II and III induction says $P\left(\eta \right)$ is true is for all $\eta \in N$