Let a,a2,...,an c with a1, a2,...,an pairwise relatively prime. Prove that if ai c for each i, then a1,a2,...an c
so far:
gcd(a1,a2,an)=1 since its pairwise then gcd (a1,an) =1, gcd (a2,an)=1
gcd(ai,aj)=1
c=ai*m
I'd suggest you to procede by induction.
Show that if $\displaystyle a_1\cdots a_k|c$, then $\displaystyle a_1\cdots a_ka_{k+1}|c$. Hint: remark that $\displaystyle a_1\cdots a_k$ and $\displaystyle a_{k+1}$ are relatively prime (this by the way could be proved by induction).
suppose a,c are relatively prime. i.e. (a,c) = 1. also, a| bc ==> a p = bc for some integer p*. so, by Euclidean algorithm, there exists x and y integers such that 1 = ax+ cy
multiplying b we get b = (ab)x + (bc)y
==> b = a(bx) + (ap)y by *
==> b = a(bx+py) .
while b,x,p,y are integers we have bx+py is an integer and so, b is an integral mulple of a.
==> a | b.
basing on this result, we go to the actual question.
a1 | c , a2 |c , a1 and a2 are reletively prime.
so, a1.a2 | c
extending this property for n integers we get a1.a2.---.an | c.
can i do it this way?
Use the recommendation of Laurent.
We just need to prove that if $\displaystyle \gcd(a,b)=1$ and $\displaystyle a|c,b|c$ then $\displaystyle ab|c$.
Since $\displaystyle a|c,b|c$ it means $\displaystyle aa'=c,bb'=c$.
Now $\displaystyle \gcd(a,b)=1\implies 1 = ax+by \implies c = acx+bcy \implies c = abb'cx+aa'bcy$.
Factor out $\displaystyle ab$ to complete the proof.
I am really lost to what your trying to do. You say to use Laurent recommendation, but you use a different proof. I know how to do the proof you did as it was required for one of our hw problems, but i dont see how solves this question of mines. Also, does my proof in the early post work?