GCD property: $b\mid ac$ implies $b\mid (a,b)(b,c)$

Question

The following is a very simple statement I want to prove:

If $a,b,c$ are non-zero integers, then $b\mid ac$ implies $b\mid (a,b)(b,c)$

Here $(a,b),[a,b]$ denote the greatest common divisor and the least common multiple between $a,b$, respectively. The symbol $\mid$ means divisibility.

Attempt: $a,b,c\mid ac$ implies $[a,b]\mid ac, [b,c]\mid ac$. Then $ac=t\frac{ab}{(a,b)}=u\frac{bc}{(b,c)}$ for some integers $t,u$.

Now, it follows that $(a,b)c=tb, (b,c)a=ub$. Multiplying we get $(a,b)(b,c)ac=tub^2$.

Since $b\mid ac$, it follows the existence of an integer $q$ s.t. $ac=bq$, so that $(a,b)(b,c)bq=tub^2$. Clearly $(a,b)(b,c)q=tub$.

Here I stuck. Can I argue something more or should I follow a different strategy?

Thank you in advance for your help.

Update I would like to avoid Bezout identity, whenever possible, because I'm interested in applications to GCD domains, where a Bezout identity does not always hold. Some answers using Bezout were posted before this update.

Answer 1

$(b,a)(b,c)= ((b,a)b,(b,a)c) = (bb,ab,bc,ac) = b(b,a,c,ac/b)$

Answer 2

Offering a different strategy (using Bezout).

Let $d_1:=\gcd(a,b)$ and $d_2:=\gcd(b,c)$. Write $ac=bk$, $d_1=au+bv$, and $d_2=bx+cy$ for $k,u,v,x,y\in\mathbb Z$. We have $$\begin{align}d_1d_2&=(au+bv)(bx+cy)\\&=abuv+(ac)uy+b^2vx+bcvy\\&=b(auv)+(bk)uy+b(bvx)+b(cvy)\\&=b(auv+kuy+bvx+cvy)\end{align}$$ and thus $b\mid d_1d_2$.

Answer 3

By Bezout: $(a,b)=ak+bl$ and $(b,c)=bm+cn$, so $(a,b)(b,c)=b\cdot \text{something} + ackn$, so if $b$ divides $ac$, it also divides LHS

Answer 4

You can write $(a,b)$ and $(b,c)$ as an intenger linear combination

$(a,b) = sa + tb$ and $(b,c) = kb + qc$ for some $s,t,q,k \in \mathbb{Z}$

Then, $(a,b)\cdot (b,c) = sakb +sacq + tbkb + tbqc = b(ask + tbk + tqc) + acsq$

But we have that $b|ac$ then $ac = bg$ for some $g \in \mathbb{Z}$

So we have that $(a,b)\cdot (b.c) = b(ask + tbk + tqc) + bgsq = b(ask + tbk + tqc + gsq)$

Thus $b|(a,b)(b,c)$

Answer 5

Here's a proof which uses the Fundamental Theorem of Arithmetic instead of the Bezout Identity, in case that holds in the places you are considering where Bezout does not. Among $a$, $b$ and $c$, there are $n$ prime factors, $p_1$ to $p_n$, for some $n \ge 0$. Also, have

$$a = \prod_{i=1}^{n} p_i^{a_i}, \; a_i \ge 0 \tag{1}\label{eq1}$$

$$b = \prod_{i=1}^{n} p_i^{b_i}, \; b_i \ge 0 \tag{2}\label{eq2}$$

$$c = \prod_{i=1}^{n} p_i^{c_i}, \; c_i \ge 0 \tag{3}\label{eq3}$$

Thus,

$$b \mid ac \text{ means } a_i + c_i \ge b_i \text{ for } 1 \le i \le n \tag{4}\label{eq4}$$

Also, you have

$$(a,b)(b,c) = \left(\prod_{i=1}^n p_i^{\min(a_i,b_i)}\right)\left(\prod_{i=1}^n p_i^{\min(b_i,c_i)}\right) = \prod_{i=1}^n p_i^{\min(a_i,b_i) + \min(b_i,c_i)} \tag{5}\label{eq5}$$

Now, for each $i$, if $\min(a_i,b_i) = b_i$ or $\min(b_i,c_i) = b_i$, then their sum would be $\ge b_i$. If, instead, $\min(a_i,b_i) = a_i$ and $\min(b_i,c_i) = c_i$, then their sum of $a_i + c_i \ge b_i$. Thus, in all cases, the exponent for $p_i$ in \eqref{eq5} is at least $b_i$, meaning $b$ divides it, i.e.,

$$b \mid (a,b)(b,c) \tag{6}\label{eq6}$$