412.1.1.8 show gcd(a',b')=1 and how would this be done in SAGE / cocalc

[karush]

412.1.1.8 show gcd(a',b')=1

$\tiny{412.1.1.8}$
$\textsf{Let $d=gcd(a,b)$
If $a=da'$ and $b=db'$, show that $gcd(a',b')=1$}$$\textsf{Let $d'=gcd(a'b')$ Since $d=gcd(a,b),$}\\
\textsf{There exists intergers s and t such that $sa+tb=d$}\\
\textsf{Therefore by substitution we have}$
$$s(da')+t(db')=d$$
$\textsf{and so}$
$$sa'+tb'=1$$
$\textsf{Since by assuption }$
$$d'|a' \textit{and} d'|b'$$
$\textsf{we have $d'|1$ Hence $d'=1$}$

ok I think this is it... typos maybe

 

[jvicens]

Great job! Your proof is correct and well-explained. Here are a few minor suggestions to improve the clarity and readability of your response:

- Consider starting your proof by stating the given information and the goal, something like: "Given $a=da'$ and $b=db'$ with $d=gcd(a,b)$, we want to show that $gcd(a',b')=1$."

- In the line where you state "Since $d=gcd(a,b)$, there exists integers s and t such that $sa+tb=d$", it might be helpful to mention that this is a well-known property of the greatest common divisor.

- When you substitute $a=da'$ and $b=db'$ into the equation $sa+tb=d$, you might want to mention that this is possible because $d=gcd(a,b)$.

- In the line where you state "Since $d'|a'$ and $d'|b'$, we have $d'|1$", you could briefly explain why this is the case. For example, you could say: "Since $d'$ is a common divisor of $a'$ and $b'$, it must also divide their linear combination $sa'+tb'=1$."

- Finally, consider ending your proof with a concluding sentence, such as: "Therefore, $gcd(a',b')=1$, as desired." This will help to clearly communicate that you have successfully proved the statement.