Greatest Common Divisor property: If $\gcd(a, b) = 1$ and $a | c$ and $b | c$, then $ab | c$

Question

Here is what I am trying to prove:

Let $a,b,c,d \in ℤ_+$ with gcd$(a,b)=1$. If $a|c$ and $b|c$, prove that $ab|c$. Does the result hold if gcd $(a,b)\neq 1$ ?

I know that gcd $(a,b)=1$ can be written $ax+by=1$ where $x,y$ are integers. And $a|c$ can be written $c=al$ where $l$ is an integer, b|c can be written $c=bk$ where $k$ is an integer, and $ab|c$ can be written $c=ab(m)$ where $m$ is an integer.

What do I do?

Answer 1

Since $ax+by=1$ hence $acx+bcy=c$. Divide throughout by $ab$ to get $cx/b$ + $cy/a$ = $c/ab$. But $b|c$ and $a|c$ hence LHS is an integer and so is RHS $\Rightarrow$ $ab|c$. The result is false if gcd$(a,b)\neq 1$. Take $a=4,b=2$ and $c=12$.

Answer 2

If a divides c, then there exists a positive integer m say, such that

$c = ma$ (1)

Likewise, b|c => that there exists $n$, such that

$c = nb$ (2)

Multiplying both sides of (1) by $b$ and both sides of (2) by $a$ gives :

$bc = mab$ (3)

$ac = nab$(4)

Subtracting both sides of (4) from those of (3):

$$(b - a)c = (m - n)ab$$

So either $ab|(b-a)$ or $ab|c$

$ab|(b-a) <=> a = b $

But $gcd(a, b) = 1$ , so this is not possible .

So therefore $ab|c$

Answer 3

Hint $\,\ a,b\mid c \iff ab\mid ac,bc\iff ab\mid \gcd(ac,bc) = \gcd(a,b)c$