I remember being in school and learning about fractions. In particular, I remember the problems we had when trying to add and subtract fractions. This problem also presented itself when we tried to multiply fractions, although we still received partial credit if we couldn’t reduce fractions to their lowest terms.
What I’m referring to is the problem of finding the Greatest Common Divisor (GCD) of two numbers. The GCD of two numbers is the largest number that divides into both numbers with a remainder of zero each time. This problem has many applications, but for most of us we can relate to it because of our frustrations with fractions.
The algorithm that I’m writing about and have written a script for is called the Euclidean Algorithm, which solves precisely this problem. To be more precise, the Euclidean Algorithm finds the greatest common divisor between two integer numbers.
There are different versions of the algorithm, but the one I have implemented finds the GCD by subtracting the smaller number from the larger number, and if the result is greater than 0, the procedure repeats itself with the two lower numbers. Otherwise, the result (the GCD) is the final number that was greater than 0.
Lets see an example:
Consider the number 9 and 21.
21 – 9 = 12. 12 is greater than 0, so we repeat the procedure with 9 and 12.
12 – 9 = 3. 3 is greater than 0, so we repeat the procedure with 3 and 9.
9 – 3 = 6. 6 is greater than 0, so we repeat the procedure with 3 and 6.
6 – 3 = 3. 3 is greater than 0, so we repeat the procedure with 3 and 3.
3 – 3 = 0. 0 is not greater than 0, so can exit the loop portion of the algorithm.
Since 3 was the last positive number that we arrived at in this procedure, we see that the GCD of 9 and 12 is 3.
To learn more and see more examples, check out My Script on The Euclidean Algorithm.
- Polynomial Arithmetic (0.661)
- Sieve of Eratosthenes (0.233)
- What is a "Hard" Problem? (0.144)
- Gaussian Elimination (0.052)