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(N/A) Let $P(n): 7^{n}-3^{n}$ is divisible by $4$.Step $1$: For $n=1$,$7^{1}-3^{1} = 4$,which is divisible by $4$. So,$P(1)$ is true.Step $2$: Assume

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(N/A) Let $P(n): 7^{n}-3^{n}$ is divisible by $4$.
Step $1$: For $n=1$,$7^{1}-3^{1} = 4$,which is divisible by $4$. So,$P(1)$ is true.
Step $2$: Assume $P(m)$ is true for some $m \in N$,i.e.,$7^{m}-3^{m} = 4k$ for some integer $k$. Thus,$7^{m} = 4k + 3^{m}$.
Step $3$: For $n=m+1$,we have $7^{m+1}-3^{m+1}$.
$= 7 \cdot 7^{m} - 3 \cdot 3^{m}$
$= 7(4k + 3^{m}) - 3 \cdot 3^{m}$
$= 28k + 7 \cdot 3^{m} - 3 \cdot 3^{m}$
$= 28k + 4 \cdot 3^{m}$
$= 4(7k + 3^{m})$.
Since $4(7k + 3^{m})$ is a multiple of $4$,$P(m+1)$ is true.
By the principle of mathematical induction,$7^{n}-3^{n}$ is divisible by $4$ for all $n \in N$.

Prove the following by using the principle of mathematical induction for all $n \in N:$
$7^{n}-3^{n}$ is divisible by $4$.

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Prove the following by using the principle of mathematical induction for all $n \in N:$$7^{n}-3^{n}$ is divisible by $4$.