00:01
Okay, so this is an easy application of bezou theorem.
00:08
This theorem tells us what? tells us that if we have two integers a and b, then we can write the greatest common divisor of a and b as, okay, let me use a prime multiplied by a, a plus b prime multiplied by b okay now we're going to use this theorem here okay so let's get started with the first part of our exercise okay we assume that c can be written as a x plus a x plus b y and we are going to show that the greatest common divisor of a and b divides c.
01:08
So we proved okay that the greatest common divisor of a and b divides c.
01:22
Well, for this part of the exercise we don't even need bazu theorem because, well, we know that this guy here that i'm going to call d, we know that d divides a and d divides b by definition of the greatest common divisor.
01:43
So this thing shows that clearly d must divide c because d dividing a and b implies that d divides a and b implies that d divides a plus by, perfect, which implies, since we have this equation here, that d divides c.
02:05
So the first part was easy.
02:08
Now for the second part, we are going to assume that the greatest common divisor of a and b divides c...