Question

$7^n-3^n$ is always divisible by

• A. 6
• B. 10
• C. 4
• D. 8

Answer 1

The correct option is C

4

Substituting n=1, we get
$7^1-3^1=4$
Let $P\left(n\right):7^n-3^n\text{is divisible by 4}$
P(1) is true
Assume P(k) is true
$7^k-3^k\text{is divisible by 4}{7}^k-3^k=4mNow,7^{k+1}-3^{k+1}=7.(7^k-3^k)+{4.3}^k=4.(7m+3^k)\to \text{divisible by 4}$
P(k+1) is true

Hence, P(n) is true.