Question

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

• A. True
• B. False

Answer 1

The correct option is A True

Consider $P\left(n\right):7^n-3^n$ is divisible by $4$

Now,

$P\left(1\right):7^1-3^1=4$

Thus, it is true for $n=1$

Let $p\left(k\right)$ is true for $n=K$

$7^k-3^{k}isdivisibleby4$

Now, prov that $P(k+1)$ is true.

$\begin{array}{c}{7}^{(k+1)}-3^{(k+1)}=7^{(k+1)}-{7.3}^k+{7.3}^k-3^{(k+1)}\\ =7(7^k-3^k)+(7-3)3^k=7\left(4d\right)+(7-3)3^k\\ =7\left(4d\right)+{4.3}^k=4(7d+3^k)\end{array}$

Hence, $P\left(n\right):7^n-3^n$ is divisible by $4$ is true.