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
gcd(a1,a2,an)=1 since its pairwise then gcd (a1,an) =1, gcd (a2,an)=1
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?