Thread: Number Theory GCD

let a,b,c in Z with (a,b)=1. If a divides c and b divides c, then ab divides c.

Hello,

This one is "simple" ^^

If a divides c, then one can write c=ka where k is an integer.
Similarly, one can write c=k'b where k' is also an integer.

Thus we have the equality ka=k'b.
So we can deduce that b divides ka.
Since b and a are coprime, b divides k. *this can be proved, see below
So one can write k=k''b where k'' is an integer.

Substitute this in c :
c=ka=k''ba

Hence ab divides c

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* proof :
let p be a divisor of b.
Since it divides b, it divides ka. But it cannot divide a otherwise gcd(a,b) wouldn't be 1. Therefore it divides k.
So for any divisor of b, it divides k. That is to say b divides k.