Question

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

• A. $7$
• B. $3$
• C. $4$
• D. $5$

Answer 1

Answer: (C)

Let $P(n):7^n-3^n$ is divisible by $4$.
For $n=1,$
$P(1):7^1-3^1=4$, which is divisible by $4$. Thus, $P(n)$ is true for $n=1.$
Let $P(k)$ be true for some natural number $k,$
i.e. $P(k):7^k-3^k$ is divisible by $4.$
We can write $7^k-3^k=4d$, where $d\in N...\left(i\right)$
Now, we wish to prove that $P(k+1)$ is true whenever P(k) is true, i.e. $7^{k+1}-3^{k+1}$ is divisible by $4.$
Now, $7^{(k+1)}-3^{(k+1)}=7^{(k+1)}-7.3^k+7.3^k-3^{(k+1)}$
$=7(7^k-3^k)+(7-3)3^k=7(4d)+4.3^k$ [using (i)]
$=4(7d+3^k)$, which is divisible by $4$.
Thus, $P(k+1)$ is true whenever $P(k)$ is true. Therefore, by the principle of mathematical induction the statement is true for every positive integer $n$.