Proving the Theorem: (b,c)=1 implies (a,bc)=(a,b)(a,c) in Number Theory

[tennesseewiz]

Homework Statement

Let a,b,c be integers. If (b,c)=1, then (a,bc)=(a,b)(a,c)

Homework Equations

This is difficult to answer because some theorems that we haven't proven yet, we can't use.

The Attempt at a Solution

Let g=(a,b) and h=(a,c), g and h are integers.
That means g|a and g|b, h|a and h|c.
That means:
a=gk, for some integer k
b=gq, for some integer q
a=hp for some integer p
c=ht for some integer t.

Multiply b and c together to get bc=(a,b)(a,c)qt. Then (a,bc)= (gk, (a,b)(a,c)qt), but that's as far as I got.

 

[matt grime]

Sometimes it is just easier to think what things mean. Suppose that d divides a and bc. Think about the prime factors of d. Can you split them into two types? (Or is uniqueness of prime factorisation something you can't use?)

 

[tennesseewiz]

Yeah, we can use uniqueness of prime factorization. We proved it last month, so that's why we can use it. I don't know how to do correct mathematical notation on the computer here, so hopefully you'll understand what I'm typing.

So d=(a,bc). d=(p1^r1)(p2^r2)(p3^r3)... or we can say d=(2^r2)(3^r3)(5^r5)... But I don't see how that's going to help me.

 

[matt grime]

So you know that gcd(a,b) is the product of the primes that divide a and b (with the right powers), similarly for gcd(a,c). And b and c are coprime so no primes divide both of them. Do you see where this is going?

 

[jkid1134]

Theorem: If (b,c)=1, then (a,bc)=(a,b)(a,c).

Let g=(a,b) and h=(a,c), g and h are integers.
That means g|a and g|b, h|a and h|c.
Therefore:
b=qg, for some integer q.
c=th for some integer t.
g|b and h|c, and (b,c)=1, so (g,h)=1.
Therefore a=pgh, for some integer p.
(a,b)=g => (a,b)/g=1 => (pgh,qg)/g=1 => g(ph,q)/g=1 => (ph,q)=1 => (p,q)=1
(a,c)=h => (a,c)/h=1 => (pgh,th)/h=1 => h(pg,t)/h=1 => (pg,t)=1 => (p,t)=1
Therefore (p,qt)=1
(a,bc)=(pgh,qgth)=gh(p,qt)=gh*1=gh=(a,b)(a,c)
Therefore (a,bc)=(a,b)(a,c)
Q.E.D.

I've never done work like this before, so I'm sure I have several form issues, but I went ahead and showed as much of my thought process as I could in hopes that you understand. To the best of my knowledge, this proves the theorem, but if I took any liberties, please, feel free to correct me.