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The Euclidean Algorithm is a classic method for calculating the greatest common divisor (gcd) of two numbers. It is based on the principle that the gcd of two numbers also divides their difference. For example, to find the gcd of two numbers, say 48 and 18, you repeatedly subtract the smaller number from the larger one and replace the larger number with the remainder until a zero remainder is obtained. The non-zero remainder then is the gcd:

• Step 1: 48 - 18 = 30 (Replace 48 with 30)
• Step 2: 30 - 18 = 12 (Replace 30 with 12)
• Step 3: 18 - 12 = 6 (Replace 18 with 6)
• Step 4: 12 - 6 = 6 (Replace 12 with 6)
• Step 5: 6 - 6 = 0 (Stop, as remainder is 0)

The gcd is 6. In modern terms, we replace subtraction with the modulus operation for efficiency, defining gcd(\(a\), \(b\)) = gcd(\(b\), \(a\) mod \(b\)), until \(b\) becomes 0. The algorithm is crucial for proving that consecutive Fibonacci numbers are relatively prime, as we will address later.

The greatest common divisor (gcd) of two non-zero integers is the largest positive integer that divides each of the integers without leaving a remainder. For instance, the gcd of 8 and 12 is 4, since 4 is the highest number that can evenly divide both 8 and 12. The concept of gcd is fundamental in various fields, like number theory, cryptography, and even in simplifying fractions. Determining the gcd is essential in demonstrating the relationship between consecutive Fibonacci numbers, where each pair is shown to be coprime, meaning their gcd is 1.

Proof by induction is a mathematical technique used to prove that a statement holds true for all natural numbers. This method involves two main steps: the base case and the inductive step. First, you show the statement is true for an initial case, usually when \(n = 1\). Then, assuming the statement holds for some arbitrary natural number \(n\), you prove it also holds for \(n+1\). If both steps succeed, the statement is proven for all natural numbers. Induction plays a vital role in recursive sequences, such as Fibonacci numbers, because you can prove properties about the entire sequence by only looking at limited consecutive elements.

Recursive sequences are a type of number series where each term after the first few is defined as a function of the preceding terms. The Fibonacci sequence is a quintessential example, where each number is the sum of the two preceding ones: \( F_{n+1} = F_{n} + F_{n-1} \) with initial terms \( F_0 = 0 \) and \( F_1 = 1 \). Recursive sequences are everywhere in mathematics, from natural phenomena like the branching patterns of trees to computer science algorithms. Understanding recursion is key to analyzing such sequences and for seeing how the properties of earlier terms influence the later ones. When it comes to Fibonacci numbers, this recursive nature facilitates the use of proof by induction to establish sophisticated properties, such as the gcd between consecutive terms, which is fundamentally linked to the Euclidean Algorithm.