Math Help - gcd question

1. gcd question

Question: Show that (ma,mb) = m(a,b) if m if greater than 0.

Answer: c = (a,b)
c = xa + yb
mc = mxa + myb = m(a,b)

This is where I'm stuck, I'm not sure how to prove that m(a,b) = (ma,mb). I also know that mc|ma and mc|mb.

Thanks!

2. Originally Posted by temp31415
Question: Show that (ma,mb) = m(a,b) if m if greater than 0.

Answer: c = (a,b)
c = xa + yb
mc = mxa + myb = m(a,b)

This is where I'm stuck, I'm not sure how to prove that m(a,b) = (ma,mb). I also know that mc|ma and mc|mb.

Thanks!
i have proved this before but i can't find the thread.. anyways..

let $d = (ma,mb)$. then $d = max + may$ for some integers x,y..
notice that, $m|max$ and $m|may$, thus, $m|d$..

so, $\frac{d}{m} = ax + by$.. hence, $\frac{d}{m} = (a,b) \implies d = m(a,b)$