Let $\displaystyle a \in \mathbb{Z} .$ with $\displaystyle a>0$. Find the greatest common divisors below.
a) $\displaystyle (a,a^n) $ where n is a positive integer.
b) $\displaystyle (3a + 5, 7a+ 12) $
The GCD of a and b, if b>a, is the same as the GCD of a and b-a. You just keep repeating that process in order to get your answer. Suppose instead I want to know $\displaystyle (2a + 3, 6a+12)$. Then we get that $\displaystyle (2a+3, 6a+12) = (2a+3, 4a+9) = (2a+3, 2a+6) = (2a+3, 3) = (2a, 3)$. You couldn't simplify that any further, without knowing a. In some cases the gcd would be 1 and in other cases it would be 3.
More generally, the gcd of x and y divides any linear combination of x and y.The GCD of a and b, if b>a, is the same as the GCD of a and b-a.
Here, $\displaystyle x=3a+5$ and $\displaystyle y=7a+12$
So it divides $\displaystyle 7x-3y$ (to eliminate a).
$\displaystyle 7x-3y=21a+35-21a-36=-1$
Since the gcd is an integer (and said to be positive), $\displaystyle gcd=1$