Problem 4 - A Gentle Introduction to the Art of Mathematics

Question

Trace through the Euclidean algorithm with inputs \(a=3731\) and \(b=2730,\) each time the assignment statement that calls the division algorithm is encountered write out the expression \(a=q b+r\).

Step-by-Step Solution

Answer

The GCD of 3731 and 2730 is 91.

Step 1 - Initial Inputs

Start with the given numbers: a = 3731 and b = 2730.

Step 2 - First Division

Perform the division of 3731 by 2730: \( 3731 = 1 \times 2730 + 1001 \). Here, the quotient q is 1, and the remainder r is 1001.

Step 3 - Update Values

Set a = 2730 and b = 1001. Move to the next iteration.

Step 4 - Second Division

Perform the division of 2730 by 1001: \( 2730 = 2 \times 1001 + 728 \). Here, the quotient q is 2, and the remainder r is 728.

Step 5 - Update Values Again

Set a = 1001 and b = 728.

Step 6 - Third Division

Perform the division of 1001 by 728: \( 1001 = 1 \times 728 + 273 \). Here, the quotient q is 1, and the remainder r is 273.

Step 7 - Continue with Updates

Set a = 728 and b = 273.

Step 8 - Fourth Division

Perform the division of 728 by 273: \( 728 = 2 \times 273 + 182 \). Here, the quotient q is 2, and the remainder r is 182.

Step 9 - Update Again

Set a = 273 and b = 182.

Step 10 - Fifth Division

Perform the division of 273 by 182: \( 273 = 1 \times 182 + 91 \). Here, the quotient q is 1, and the remainder r is 91.

Step 11 - Update Values

Set a = 182 and b = 91.

Step 12 - Sixth Division

Perform the division of 182 by 91: \( 182 = 2 \times 91 + 0 \). Here, the quotient q is 2, and the remainder r is 0.

Step 13 - Conclusion

Since the remainder is now 0, the algorithm terminates. The greatest common divisor (GCD) of 3731 and 2730 is the last non-zero remainder, which is 91.

Key Concepts

Greatest Common Divisor

The greatest common divisor, or GCD, is the largest positive integer that divides two numbers without leaving a remainder. In simpler terms, it's the biggest number that two or more numbers share as a factor. For example, the GCD of 8 and 12 is 4, because 4 is the largest number that can evenly divide both 8 and 12.

The GCD is useful in various mathematical applications, like simplifying fractions and number theory. One of the most efficient ways to find the GCD of two numbers is by using the Euclidean algorithm, which we will cover next.

Division Algorithm

The division algorithm is a fundamental concept in mathematics used in the Euclidean algorithm to find the GCD. It states that for any two integers, a and b, where a is the dividend and b is the divisor, there exist unique integers q (quotient) and r (remainder) such that:
where 0 ≤ r < b.
This means you can break down a division into a quotient and a remainder. For example, dividing 3731 by 2730, we get a quotient (q) of 1 and a remainder (r) of 1001, because:
3731 = 1 * 2730 + 1001.

The division algorithm is repeatedly applied in the Euclidean algorithm to reduce the size of the numbers and, ultimately, find the GCD.

Step-by-Step Mathematical Process

The Euclidean algorithm is a step-by-step mathematical process used to find the GCD of two numbers. Let's go through the steps using the initial exercise:

• Step 1: Initial Inputs. Start with a=3731 and b=2730.
• Step 2: First Division. Perform the division 3731 by 2730: 3731 = 1×2730 + 1001. The remainder r is 1001.
• Step 3: Update Values. Set a=2730 and b=1001. Move to the next iteration.
• Step 4: Second Division. Perform the division 2730 by 1001: 2730=2×1001 + 728. The remainder r is 728.
• Step 5: Update Values Again. Set a=1001 and b=728.
• Step 6: Third Division. Perform the division 1001 by 728: 1001=1×728 + 273. The remainder r is 273.
• Step 7: Continue with Updates. Set a=728 and b=273.
• Step 8: Fourth Division. Perform the division 728 by 273: 728=2×273 + 182. The remainder r is 182.
• Step 9: Update Again. Set a=273 and b=182.
• Step 10: Fifth Division. Perform the division 273 by 182: 273=1×182 + 91. The remainder r is 91.
• Step 11: Update Values. Set a=182 and b=91.
• Step 12: Sixth Division. Perform the division 182 by 91: 182=2×91 + 0. The remainder r is 0.
• Step 13: Conclusion. Since the remainder is now 0, the process ends. The last non-zero remainder, 91, is the GCD of 3731 and 2730.

This method ensures that you can find the GCD efficiently and accurately by repeatedly applying the division algorithm until the remainder is zero.