An Example of GCD, and Extended Euclidean Algorithm In Finding the Bezout Coefficients 

it is not easy to explain the details with Shorts. I will consider a longer video on the Extended Euclidean algorithm. For now, please refer this wiki page en.wikipedia.org/wiki/Extended_Euclidean_algorithm

*Euclidean Algorithm (For GCD):* 🌟 🤯 Another efficient way to find the greatest common divisor (and thus derive common factors) is using the Euclidean algorithm: Step 1: Divide the larger number by the smaller one and take note of the *remainder.* Step 2: Replace the larger number with the smaller one and repeat this process until you reach a remainder of zero. Step 3: The last non-zero remainder will be your GCD. *For example:* For numbers 12 and 18: “18 ÷ 12 = 1 remainder *6* ” {12 * 1 = 12 (so this is why the 1 is there. 👆) and 12 + 6 = 18 (so this is why the 6 is there. 👆)} [If using calculator *Do 18 divided by 12 so 18/12 is 1.5,* *next round the number down to have NO DECIMALS so 1.5 becomes 1,* *then multiply that whole number to 12 so 1 times 12 is 12,* *lastly subtract 18 by 12 so 18 - 12 is 6 the answer.* ] “12 ÷ 6 = 2 remainder 0” Thus, GCD (12,18) = *6* , confirming our earlier findings. ✔️ 👍

*”Steps to Use the Euclidean Algorithm with a Calculator”* *Step 1: Divide the Larger Number by the Smaller Number* • Input the larger number (18) divided by the smaller number (12) into your calculator. • Record the integer part of the quotient. For example, 18 ÷ 12 = 1.5, so take 1. *Step 2: Multiply and Subtract* • Multiply the integer part from Step 1 by the smaller number: 1 × 12 = 12. • Subtract this result from the larger number to find the remainder: 18 − 12 = 6. *Step 3: Replace and Repeat* • Now replace the larger number with the smaller number (12) and use the remainder (6) as your new smaller number. • Repeat Step 1: Calculate 12 ÷ 6 = 2. The integer part is 2. *Step 4: Final Calculation* • Multiply again: 2 × 6 = 12. • Subtract to find a new remainder: 12 − 12 = 0. *Conclusion* • When you reach a remainder of zero, the last non-zero remainder is your GCD. In this case, GCD(12,18) = *6*

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