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

Printable View

- Jul 19th 2008, 06:46 PMJCIRNumber Theory GCD
let a,b,c in Z with (a,b)=1. If a divides c and b divides c, then ab divides c.

- Jul 20th 2008, 09:04 AMMoo
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 (Evilgrin)

----------------------------

* 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.