Problem 56 - Discrete Mathematics with Applications

Question

Let \(a\) and \(b\) be any two positive integers with \(a \geq b\). Using the sequence of equations in the euclidean algorithm prove that \(\operatorname{gcd}\\{a, b\\}=\operatorname{gcd}\left\\{r_{n-1}, r_{n}\right\\}, n \geq 1\).

Step-by-Step Solution

Answer

In summary, we can prove gcd(a, b) = gcd(rₙ₋₁, rₙ) for all n ≥ 1 using mathematical induction on the remainders from the Euclidean algorithm, where a = b * q₁ + r₁, b = r₁ * q₂ + r₂, ..., rₙ₋₂ = rₙ₋₁ * qₙ + rₙ (with rₙ ≠ 0), and rₙ₋₁ = rₙ * qₙ₊₁ (with remainder 0). The base case corresponds to n = 1, and our induction step demonstrates that if the statement is true for n = k, then it is also true for n = k + 1. This proves that gcd(a, b) = gcd(rₙ₋₁, rₙ) for all n ≥ 1.

Step 1: Understanding the Euclidean Algorithm

The Euclidean algorithm is an efficient method for computing the gcd of two positive integers. Given two positive integers a and b (with a ≥ b), the algorithm is as follows: 1. Divide a by b and get the remainder r₁ (a = b*q₁ + r₁, where q₁ is the quotient). 2. If r₁ = 0, then gcd(a, b) = b. 3. Else, replace a with b and b with r₁, and repeat the process until the remainder is 0.

Step 2: Applying the Euclidean Algorithm

Let's apply the algorithm to get the sequence of remainders: 1. a = b * q₁ + r₁ 2. b = r₁ * q₂ + r₂ 3. r₁ = r₂ * q₃ + r₃ ... n. rₙ₋₂ = rₙ₋₁ * qₙ + rₙ (with rₙ ≠ 0) n+1. rₙ₋₁ = rₙ * qₙ₊₁ (with remainder 0) The Euclidean algorithm will stop at the (n+1)th step since the remainder rₙ is zero.

Step 3: Prove gcd(a, b) = gcd(rₙ₋₁, rₙ)

We would use mathematical induction on the remainders to prove this: Base case, n = 1: We need to prove gcd(a, b) = gcd(b, r₁). According to Step 2, we have a = b * q₁ + r₁. Since a is a multiple of gcd(a, b) and a - b * q₁ = r₁, then r₁ must also be a multiple of gcd(a, b). So gcd(a, b) must divide both b and r₁, and hence gcd(a, b) ≤ gcd(b, r₁). Furthermore, since gcd(b, r₁) divides both b and r₁, it must also divide a, so gcd(b, r₁) ≤ gcd(a, b). Combining these inequalities, we have gcd(a, b) = gcd(b, r₁). Induction step: Suppose the statement is true for n = k, i.e., gcd(rₖ₋₁, rₖ) = gcd(a, b). We must now prove it for n = k + 1, i.e., gcd(rₖ, rₖ₊₁) = gcd(a, b). For n = k + 1, we have rₖ = rₖ₊₁ * qₖ₊₁ + rₖ₊₂ with remainder 0. So, gcd(rₖ₋₁, rₖ) = gcd(rₖ, rₖ₊₁). Since gcd(rₖ₋₁, rₖ) = gcd(a, b) (from our induction hypothesis), we get gcd(a, b) = gcd(rₖ, rₖ₊₁). Hence, by the principle of mathematical induction, we have proven that gcd(a, b) = gcd(rₙ₋₁, rₙ) for all n ≥ 1.

Key Concepts

gcd (greatest common divisor)

The greatest common divisor (gcd) of two numbers is the largest number that divides both of them without leaving a remainder. It's a fundamental concept in number theory and an essential tool for simplifying fractions and solving problems involving divisibility.

For example, to find the gcd of 48 and 18, we can list their divisors: For 48, they are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48, and for 18, they are 1, 2, 3, 6, 9, and 18. The largest number appearing in both lists is 6, which is the gcd(48, 18).

However, listing divisors becomes impractical for larger numbers, which is where the Euclidean algorithm comes in handy. It provides a systematic way to calculate the gcd by using a series of divisions and focusing on remainders, a process that is both efficient and powerful for dealing with large integers.

mathematical induction

Mathematical induction is a proof technique that is used to establish the truth of an infinite number of statements. It works by first proving that a base case is true, then assuming a general case or 'inductive hypothesis' for some integer 'k', and using this to show that the next case, 'k+1', is also true.

The principle of mathematical induction is akin to a domino effect; knocking the first domino (the base case) should, in theory, knock down each subsequent domino, establishing the truth of the statement for all natural numbers.

Steps of Mathematical Induction

• Base Case: Verify that the statement holds for the initial value, often n=1.
• Inductive Hypothesis: Assume the statement holds for some arbitrary positive integer k.
• Inductive Step: Under the inductive hypothesis, prove that the statement holds for k+1.

Using induction to establish the equality of gcds at each step of the Euclidean algorithm demonstrates the recursive nature of this algorithm.

remainder sequence

In the context of the Euclidean algorithm, the remainder sequence is the list of remainders obtained at each step of the algorithm when the previous remainder is divided by the current one. It is a crucial component of the algorithm because it tells us when the process ends: when we reach a remainder of 0. The sequence of remainders starts with the two numbers we want to find the gcd of, traditionally called 'a' and 'b', and each subsequent number in the sequence is strictly smaller than the previous one.

This decreasing nature of the remainder sequence is what guarantees that the Euclidean algorithm will eventually terminate, since we cannot have an infinite strictly decreasing sequence of positive integers. Once the remainder is 0, the gcd is the number we divided by to get that 0 remainder.

Example of a Remainder Sequence

If we start with 56 and 32, our remainder sequence might look like this: 56, 32, 24, 8, 0. The last non-zero remainder, 8, is the gcd of 56 and 32.