Question

For each of the following pairs of integers, find their greatest common divisor using the Euclidean Algorithm: (i) 34,21 : (ii) 136,51 : (iii) 481,325 ; (iv) 8771,3206 .

Short answer

(i) GCD is 1; (ii) GCD is 17; (iii) GCD is 13; (iv) GCD is 1.

Step-by-step solution

Understanding the Euclidean Algorithm

The Euclidean Algorithm finds the greatest common divisor (GCD) of two integers by repeatedly applying the division algorithm until the remainder is zero. The basic idea is that \( \text{GCD}(a, b) = \text{GCD}(b, a \mod b) \) where \(a \mod b\) is the remainder when \(a\) is divided by \(b\).

Pair (i): 34 and 21

1. Compute \(34 \mod 21\): The remainder when 34 is divided by 21 is 13. 2. Compute \(21 \mod 13\): The remainder is 8. 3. Compute \(13 \mod 8\): The remainder is 5.4. Compute \(8 \mod 5\): The remainder is 3.5. Compute \(5 \mod 3\): The remainder is 2.6. Compute \(3 \mod 2\): The remainder is 1.7. Compute \(2 \mod 1\): The remainder is 0.The GCD is the last non-zero remainder, which is 1.

Pair (ii): 136 and 51

1. Compute \(136 \mod 51\): The remainder is 34.2. Compute \(51 \mod 34\): The remainder is 17.3. Compute \(34 \mod 17\): The remainder is 0.The GCD is the last non-zero remainder, which is 17.

Pair (iii): 481 and 325

1. Compute \(481 \mod 325\): The remainder is 156.2. Compute \(325 \mod 156\): The remainder is 13.3. Compute \(156 \mod 13\): The remainder is 0.The GCD is the last non-zero remainder, which is 13.

Pair (iv): 8771 and 3206

1. Compute \(8771 \mod 3206\): The remainder is 2359.2. Compute \(3206 \mod 2359\): The remainder is 847.3. Compute \(2359 \mod 847\): The remainder is 665.4. Compute \(847 \mod 665\): The remainder is 182.5. Compute \(665 \mod 182\): The remainder is 19.6. Compute \(182 \mod 19\): The remainder is 11.7. Compute \(19 \mod 11\): The remainder is 8.8. Compute \(11 \mod 8\): The remainder is 3.9. Compute \(8 \mod 3\): The remainder is 2.10. Compute \(3 \mod 2\): The remainder is 1.11. Compute \(2 \mod 1\): The remainder is 0.The GCD is the last non-zero remainder, which is 1.

Key concepts

Greatest Common Divisor

The Greatest Common Divisor, commonly abbreviated to GCD, is the largest positive integer that divides two or more integers without leaving a remainder. For example, for the numbers 12 and 15, the GCD is 3. Finding the GCD is important in simplifying fractions and is a fundamental concept in number theory. It's especially useful when solving various problems in mathematics and engineering.

The Euclidean Algorithm is a systematic method of finding the GCD of two integers. By using repeated division, it provides an efficient way to reach the correct answer without the need to factor multiplicatively the integers fully.

• The GCD of two numbers remains the same if the larger number is replaced by its difference with the smaller number.
• The algorithm continues until the remainder becomes zero.
• The last non-zero remainder is the GCD of the original pair of numbers.

Division Algorithm

The Division Algorithm is a theorem that guarantees the division of one integer by another, producing a quotient and a remainder. It states that for any integers a and b, with b > 0, there exist unique integers q (quotient) and r (remainder) such that:

\[ a = bq + r \]

where \( 0 \leq r < b \). This formula is the backbone of the Euclidean Algorithm, simplifying the process of finding the GCD. It ensures that each step in the Euclidean Algorithm reduces the problem size, drawing closer to the solution.

• It's useful in applications where dividing tasks into smaller parts is necessary.
• Helps in understanding the behavior of remainders, critical in modular arithmetic.

The Division Algorithm's simplicity makes it a very effective tool for solving problems related to number theory, encryption, and even computer science.

Integer Pairs

An integer pair refers to any two whole numbers, positive or negative. Working with integer pairs is an essential skill in mathematics, as it supports various computations, including finding the GCD.

When applying the Euclidean Algorithm to integer pairs, the problem becomes an exercise in dividing one number by another and utilizing the remainder to form a new pair until the solution is found. This iterative process highlights the simplicity and power of reducing larger problems into more manageable pieces.

• Pairs make up the building blocks for operations, sequences, and series in math.
• Finding relationships between pairs is crucial for solving equations and inequalities.

Understanding how integer pairs interact through division and remainders facilitates deeper insights into more complex mathematical procedures.

Mathematics Education

Mathematics Education focuses on teaching and learning mathematics in a way that encourages comprehension, problem-solving skills, and practical application. Approaches like the Euclidean Algorithm serve as excellent teaching tools because they provide hands-on practice with clear, applicable results.

Such algorithms help students develop logical reasoning and analytical skills. Additionally, understanding fundamental concepts like the GCD or the Division Algorithm prepares students for more advanced topics in mathematics.

• Enhances the ability to think critically about problems.
• Offers a structured pathway from basic arithmetic to complex analysis.
• Encourages the ongoing learning and application of mathematics in various fields.

Incorporating algorithms that are both historical and practical helps students appreciate the depth and utility of mathematics, making it more relatable and interesting.