1.5 Algebra Number Theory

Let \( F_n \) denote the \( n \)-th Fibonacci number, defined by \( F_1=1, F_2=1 \), and \( F_{k+2} = F_{k+1} + F_k \) for \( k \ge 1 \).
Prove by induction that for all positive integers \( n \), \( \gcd(F_n, F_{n+1}) = 1 \).

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