00:01
Hello everyone, in this question let us take a and b be non -zero integers as per the equation.
00:16
Now let us move on to the first part of the equation.
00:19
If c is an integer such that there exists x and y with ax plus by which is equal to c then we need to prove that gcd of ab divides c.
00:30
So now let us take d to be the gcd of ab which implies that d divides a and d divides b.
00:47
This is by the definition of algorithm of these gcd since we are given that ax plus by which is equal to c so which implies that d also divides c.
01:07
Since d divides a, d divides b, d divides which implies that d divides ax and d divides by so which in turn divides d divides c that is by d divides ax plus by.
01:29
So this is the proof for the first part.
01:31
Now let us move on to the next part.
01:33
Let us take t to be the gcd of ab which implies that d divides a and d divides b.
01:47
So now if there exists integers x and y such that ax plus by to be equal to 1 now we need to explain why the gcd of ab is 1.
02:00
So since here ax plus by which is equal to 1 which implies that since d divides a and d divides b and d divides ax plus by which implies that d divides 1 which implies that d is equal to 1.
02:32
So hence we have proved that gcd of ab is 1...