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

Let $P(n)$ be the statement: $7^n-3^n$ is divisible by $4$.

Base case: For $n=1$,

which is divisible by $4$. So $P(1)$ is true.

Inductive step: Assume $P(k)$ is true for some $k\ge1$, i.e. there exists an integer $m$ such that

We must show $7^{k+1}-3^{k+1}$ is divisible by $4$.

Write $7\cdot7^k = 7(7^k-3^k) + 7\cdot3^k$, so

Substituting the induction hypothesis $7^k-3^k=4m$: …

Divisibility induction. Write $7^{k+1}-3^{k+1}$ as $7(7^k-3^k)+4\cdot3^k$ so that the induction hypothesis $4\mid(7^k-3^k)$ can be substituted …

• Writing $7^{k+1}-3^{k+1}=7(7^k-3^k)$ without the correction term $4\cdot3^k$ — an algebra error.
• Stating the hypothesis as "$7^k-3^k=4$" instead of "$7^k-3^k=4m$ for some integer $m$". …