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Let $a$ and $b$ be two integers such that $\left(a,b\right) = 1$. Prove that $\left(a+b, ab\right) = 1$.

$(a,b)=1$ means $a$ and $b$ have no prime factors in common

$ab$ is simply the product of factors of $a$ and factors of $b$.

Let's say $k\mid a+b$ where $k$ is some factor of $a$.

Then $ka=a+b$ and $ka-a=b$ and $a(k-l)=b$.

So $a(k-l)=b, \ a\mid a(k-1)$ [$a$ divides the left hand side] therefore $a\mid b$ [the right hand side].

But $(a,b)=1$ so $a$ cannot divide $b$.

We have a similar argument for $b$.

So $a+b$ is not divisible by any factors of $ab$.

Therefore, $(a+b, ab)=1$.

Would this be correct? Am I missing anything?

14 Answers 14