Question

For finding the greatest common divisor of two integers. A method based on the division algorithm is used called

Answer 1

Hint: First, we will see the greatest common divisors, which are the comparison of two or more than two numbers or integers that has a common divisor, and also of these common divisors, we have to find which number is the greatest.
For example: $\gcd (6,12)$we have to find the greatest common divisors of the numbers six and twelve.
First, we will find the common divisors of the numbers six and twelve, which are $1,2,3,6$.
Since there are four common divisors for the six and twelve. Now we will check which among the terms are greatest, thus we get $\gcd (6,12) = 6$ which is the number common and also greatest among all numbers. Hence this is called greatest common divisors.

Complete step-by-step solution:
Let us find the greatest common divisors of the two or more than two (we cannot find the greatest common divisors of a single number because a single number itself is the greatest common divisor).
First, we have to find the dividing terms of the given numbers and then we will check which among the numbers are greatest from the smallest remainder.
Hence this greatest common divisor method is the process of the division only, the difference is we check for the greatest divisor’s terms.
Euclid was the one who found this greatest common division method and later these expressions are used in the division algorithm too.
Therefore, the Euclid algorithm is $\gcd (a,b)$ and later it is used for the division algorithm $a = bq + r$ where a and b are the integers and q and r are the quotient and remainder respectively.
Hence the method is called Euclid’s division algorithm.

Note: Since Euclid uses only the grease common divisors for his method to division algorithm.
There are also some other methods like LCM which is the least common multiple and also HCF's highest common factor is more likely the greatest common f=division.
Also, if $\gcd (a,b) = 1$ then it is called the relatively prime numbers as their common divisors are number one only.
Example: $\gcd (2,3) = 1$