1. relatively prime integers....

Hello everbody...

Let b and c be relatively prime integers, and suppose a is an integer that is
divisible by both b and c. Prove that bc|a.

2. If a is divisible by both b and c, we have:

$\displaystyle bk = a$

$\displaystyle cj = a$

$\displaystyle b = a/k$

$\displaystyle c = a/j$

Therefore:

$\displaystyle bc = a/jk$

$\displaystyle jk = i$

$\displaystyle bc(i) = a$

Thus:

$\displaystyle bc | a$

3. Originally Posted by Aryth
If a is divisible by both b and c, we have:

$\displaystyle bk = a$

$\displaystyle cj = a$

$\displaystyle b = a/k$

$\displaystyle c = a/j$

Therefore:

***$\displaystyle bc = a/jk$***

$\displaystyle jk = i$

$\displaystyle bc(i) = a$

Thus:

$\displaystyle bc | a$
ummm....$\displaystyle bc = a^2/jk$. how does that work?

4. b and c divide a $\displaystyle \Rightarrow$ $\displaystyle a=hb=kc$ for some integers h and k.

b and c are relatively prime means there are integers $\displaystyle p,q$ such that $\displaystyle pb+qc=1$.

Now multiply through by a.

$\displaystyle pba+qca=a$ $\displaystyle \Rightarrow$ $\displaystyle a=pb(kc)+qc(hb)=(pk+qh)bc$.