I'm trying to understand 100% intuitively and rigorously ( at the same time ) almost all facts in basic number theory. I'm going really slow-paced and at the moment i didn't reach primes and unique factorization. I was able to gain full intution and understanding ( while also knowing how to prove ) about basic properties of divisibility relation ( poset with Z* ), euclidean division, euclidean algorithm, bezout identity, the fact that every common divisor divides the gcd, the "linearity" of gcd ( gcd(ma,mb) = m.gcd(a,b) ) .
But now there are some theorems that i'm struggling to 100% intuitively understand.
For instance, the fact that if gcd(a,d) =1 and d | ab, then d | b.
I know how to formally prove that.
If gcd(a,d) =1 is the least positive linear combination of a and d, and this is obtained with some pair (x,y), then ax + dy = 1.
But while (x,y) is the "minimal pair" for a and y and gives gdc(a,b) as result, it is also a "minimal pair" for ab and db and gives b.gcd(a,b) as result. So gcd(ab,db) = b, and if the theorem is saying that d is a common divisor of ab and db ( because it divides ab, and clearly divides db ) , then it must divides the gcd of (ab,db), that is, b.
Or more shortly, there exists x,y s.t ax + dy = 1 , which implies
bax + dby = b . If d | ab, since it divides bd, it has to divide b.
But i want to get to the point where i see intuitively why this is true, not having to go into the mentioned proof. From that proof, i can only get that if gcd(a,d)=1, common divisors of the b-multiple of (a,d) , d in the case of the theorem, must also divide the gcd of the b-multiple of (a,d) , that is, must divide b.
But i can't get past that level to increase my intuition. I still don't know why if some d divides ab, and it doesnt divide a ( gcd(a,d)=1 ) , it has to divide b.
Is there any way, point of view or anything essential that doesnt use unique factorization and primes that i'm not seeing that can be considered when thinking about this theorem , maybe some intuitive corollaries of the topics i said i understood, that enables us to understand the truth of this theorem ( and related theorems as well ) autoamtically and intuitively ?
But if there isn't any way to see that more intuitively withouth using unique prime factorization and primes, then please to use it.
Thanks a lot in advance !