Question

For every positive integer $n$, prove that $7^n-3^n$ is divisible by $4$.

Answer 1

Step $\left(1\right):$ Assume given statement
Let the given statement be $P\left(n\right)$, i.e.
$P\left(n\right):7^n-3^n$ is divisible by $4$.

Step $\left(2\right):$ Checking statement $P\left(n\right)$ for $n=1$
Put $n=1$ in $P\left(n\right)$, we get
$=7^1-3^1=7-3=4$ which is divisible by $4$.
Thus, $P\left(n\right)$ is true for $n=1$.

Step $\left(3\right):$ $P\left(n\right)$ for $n=K$
Put $n=K$ in $P\left(n\right)$, and assume that $P\left(K\right)$ is true for some
natural number $K$ i.e.
$P\left(K\right)=7^K-3^K$ is divisible by $4$.
We can write
$7^K-3^K=4d$, where $d\in N\cdots \left(1\right)$

Step $\left(4\right):$ Checking statement $P\left(n\right)$ for $n=K+1$
Now, we shall prove that $P(K+1)$ is true whenever $P\left(K\right)$ is true.We have
$7^{K+1}-3^{K+1}$
$=7^{K+1}-7\cdot 3^K+7\cdot 3^K-3^{K+1}$
$=7(7^K-3^K)+3^K(7-3)$
$=7\left(4d\right)+4\cdot 3^K$
$=4(7d+3^K)$
From the last line, we can see that
$7^{K+1}-3^{K+1}$ is divisible by $4$.
Thus, $P(K+1)$ is true when $P\left(K\right)$ is true.
Final Answer:
Therefore, by principle of Mathematical Induction, the statement $P\left(n\right)$ is true for every positive integer $n$.