3. (30%) (a) Assume $N$ is a positive integer, please show that the statement "if $2^N - 1$ is prime, then $N$ is prime" is true using proof by contrapositive. (10%)

(b) $5n + 7m = N$, where $n, m$ are integers and $n \ge 0$, $m \ge 0$, $N \ge 24$. Please show that we can always find $n, m$ to satisfy the equality. For example, $(n,m) = (2,2)$ corresponds to $5 \cdot 2 + 7 \cdot 2 = 24$ and $(n,m) = (5,0)$ corresponds to $5 \cdot 5 + 7 \cdot 0 = 25$. (10%)