+31 Hi, I was recently learning about the euclidean algorithm when I found a proof which confused me.
Until recently, I had thought the notation a | b meant a was a factor of b, in that order only.
But the proof explained that for a number g which was the gcd of (a, b-a), a and b-a were both divisible by g, and so the sum (b) was also divisible by g.
This resulted in the conclusion that g | gcd(a,b).
The proof also came with a converse: g was now the gcd of (a, b), causing a and b to be divisible by g which meant their difference(b-a) was also divisible.
This resulted in the conclusion that gcd(a,b) | gcd(a, b-a).
Lining these two conclusions up invalidates the definition that a | b means only a is a factor of b, as unless a and b are equal, both the equations a | b (gcd(a,b) | gcd(a, b-a)) and b | a (gcd(a, b-a) | gcd(a, b)) cannot be true. It must be one or the other.
So, my question is: Does a | b mean that either a is a factor of b or b is a factor of a? or is there a specific order?
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** Edited by Melody to make it more readable.
