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) 1; (ii) 17; (iii) 13; (iv) 7.

Step-by-step solution

Understand the Euclidean Algorithm

The Euclidean Algorithm is used to find the greatest common divisor (GCD) of two integers. It involves repeated division where the remainder becomes the new divisor until a remainder of zero is achieved. The last non-zero remainder is the GCD.

Apply Euclidean Algorithm to 34 and 21

1. Divide 34 by 21, which gives a quotient of 1 and a remainder of 13. 2. Now divide 21 by 13. This gives a quotient of 1 and a remainder of 8. 3. Divide 13 by 8. This results in a quotient of 1 and a remainder of 5. 4. Divide 8 by 5. This yields a quotient of 1 and a remainder of 3. 5. Divide 5 by 3. This gives a quotient of 1 and a remainder of 2. 6. Divide 3 by 2. This gives a quotient of 1 and a remainder of 1. 7. Divide 2 by 1. The remainder is 0, so the GCD is 1.

Apply Euclidean Algorithm to 136 and 51

1. Divide 136 by 51, which gives a quotient of 2 and a remainder of 34. 2. Divide 51 by 34. This gives a quotient of 1 and a remainder of 17. 3. Divide 34 by 17. The remainder is 0. Thus, the GCD is 17.

Apply Euclidean Algorithm to 481 and 325

1. Divide 481 by 325, which gives a quotient of 1 and a remainder of 156. 2. Divide 325 by 156. This results in a quotient of 2 and a remainder of 13. 3. Divide 156 by 13. The remainder is 0, so the GCD is 13.

Apply Euclidean Algorithm to 8771 and 3206

1. Divide 8771 by 3206, which gives a quotient of 2 and a remainder of 2359. 2. Divide 3206 by 2359. This gives a quotient of 1 and a remainder of 847. 3. Divide 2359 by 847. The quotient is 2 and the remainder is 665. 4. Divide 847 by 665. The quotient is 1 and the remainder is 182. 5. Divide 665 by 182. The quotient is 3 and the remainder is 119. 6. Divide 182 by 119. The quotient is 1 and the remainder is 63. 7. Divide 119 by 63. The quotient is 1 and the remainder is 56. 8. Divide 63 by 56. The quotient is 1 and the remainder is 7. 9. Divide 56 by 7. The remainder is 0, so the GCD is 7.

Key concepts

Greatest Common Divisor

The concept of the greatest common divisor (GCD) is essential in understanding relationships between numbers in mathematics. The GCD of two or more integers is the largest positive integer that divides all of them without leaving a remainder. Finding the GCD can help in simplifying fractions, solving Diophantine equations, and is foundational in number theory.

The Euclidean Algorithm is a structured method to find the GCD of two numbers. It simplifies the process by using division and remainder calculations repeatedly until a remainder of zero is achieved. The last non-zero remainder is then the GCD of the original two integers.

For example, consider the numbers 34 and 21. By following the Euclidean Algorithm:

• 34 divided by 21 = 1 remainder 13
• 21 divided by 13 = 1 remainder 8
• 13 divided by 8 = 1 remainder 5
• 8 divided by 5 = 1 remainder 3
• 5 divided by 3 = 1 remainder 2
• 3 divided by 2 = 1 remainder 1
• 2 divided by 1 = 2 remainder 0

Here, the process stops with a remainder of zero and leaves us with 1 as the GCD.

Integer Division

Integer division is a fundamental math concept where one integer (the dividend) is divided by another (the divisor) to produce a quotient and a possible remainder. Unlike floating-point division that results in decimals, integer division only produces whole numbers.

In the context of the Euclidean Algorithm, integer division plays a crucial role. Each step in the algorithm involves dividing the current dividend by the divisor to find a whole number quotient and a remainder. This remainder then becomes the new divisor.

Consider the example with integers 136 and 51:

• 136 divided by 51 gives a quotient of 2 and a remainder of 34
• 51 divided by 34 results in a quotient of 1 and a remainder of 17
• 34 divided by 17 gives a remainder of 0

Understanding integer division is vital for effectively using the Euclidean Algorithm to find the greatest common divisor, as it relies on finding whole number divisions at each stage.

Mathematics Education

Mathematics education provides the knowledge and skills needed to understand and apply various math concepts, including integer division and the Euclidean Algorithm, both of which are integral in learning about the greatest common divisor.

Effective math education employs hands-on exercises, like finding the GCD using real numbers, to solidify understanding. This progressive approach helps students develop problem-solving skills and mathematical reasoning.

Teaching students through exercises like calculating the GCD with numbers such as 481 and 325, or 8771 and 3206, not only teaches them algorithms and division but also highlights the interconnectedness of mathematical concepts. Activities like these help bridge the gap between theoretical math and its practical applications, helping students become proficient in mathematics.