How to prove the Eulicides algorithm?

Question

Mathematics

The task is next. Prove the Eulicides algorithm.
m = qn+r, where q is some number; m - First number; n - Second number; r - The remainder of dividing m by n.
If r = 0, then m is a multiple of n, so n is GCD. (This is clear)
If r != 0, then any divisor of both m and n must be a divisor m - qn = r, and any divisor of n and r must also be a divisor qn + r = m. Thus, the set of divisors of the number {m, n}coincides with the set of divisors of {n, r}. Therefore, the pairs of numbers {m, n}and {n, r}have the same gcd.
The question is, what does "a certain number 'q' mean, because it is not in the algorithm? And how does this proof happen in general? I
sorted through real numbers, but I'm not sure if I'm thinking in the right direction.

Answer 1

Writing m \u003d qn + r means that when dividing m by n , we get the quotient q and the remainder r .