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.

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- Mar 16th 2008, 09:30 AMpizzaRusher1234relatively 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. - Mar 16th 2008, 11:58 AMAryth
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:

$\displaystyle bc | a$ - Mar 16th 2008, 12:37 PMpizzaRusher1234
- Mar 16th 2008, 01:00 PMJaneBennet
*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*.

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