Prove that $(ma, mb) = |m|(a, b)\ $ [GCD & LCM Distributive Law]

Below are sketches of six proofs of the GCD Distributive Law $\rm\:(ax,bx) = (a,b)x\:$ by various approaches: Bezout's identity, the universal gcd property, unique factorization, and induction.

First we show that the gcd distributive law follows immediately from the fact that, by Bezout, the gcd may be specified by linear equations. Distributivity follows because such linear equations are preserved by scalings. Namely, for naturals $\rm\:a,b,c,x \ne 0$

$\rm\qquad\qquad \phantom{ \iff }\ \ \ \:\! c = (a,b) $

$\rm\qquad\qquad \iff\ \: c\:\ |\ \:a,\:b\ \ \ \ \ \ \&\ \ \ \ c\ =\ na\: +\: kb,\ \ \ $ some $\rm\:n,k\in \mathbb Z$

$\rm\qquad\qquad \iff\ cx\ |\ ax,bx\ \ \ \&\ \ \ cx = nax + kbx,\ \,$ some $\rm\:n,k\in \mathbb Z$

$\rm\qquad\qquad { \iff }\ \ cx = (ax,bx) $

Readers familiar with ideals may note that these equivalences are captured more concisely in the distributive law for ideal multiplication $\rm\:(a,b)(x) = (ax,bx),\:$ when interpreted in a PID or Bezout domain, where the ideal $\rm\:(a,b) = (c)\iff c \approx gcd(a,b)$. This proof fails in non-Bezout domains like $\,\Bbb Z[x]\,$ and $\,\Bbb Q[x,y],\,$ e.g. $\,(x,2) = 1 = (x,y)\,$ but the gcds cannot be written as linear combinations, else $\, 1 = x\,f(x,y) + y\,g(x,y)\Rightarrow\,1=0\,$ by evaluating at $\,x=0=y.\,$ But the next proof works fine there there (or in any gcd domains).

Alternatively, more generally, in any integral domain $\rm\:D\:$ we have

Theorem $\rm\ \ (a,b)\ \approx\ (ax,bx)/x\ \ $ if $\rm\ (ax,bx)\ $ exists in $\rm\:D\ \,$ [$c\approx d := c,d\,$ associate: $\,c\mid d\mid c$]

Proof $\rm\quad\! c\mid(a,b)\!\!\overset{\rm def\!\!}\iff c\ |\ a,b \iff cx\ |\ ax,bx \!\!\overset{\rm def\!\!}\iff cx\ |\ (ax,bx) \iff c\ |\ (ax,bx)/x$

where $\,\rm x\mid ax,bx\ \smash{\overset{\rm def}\Rightarrow}\ x\mid (ax,bx).\,$ Thus $\rm\,(a,b) \approx (ax,bx)/x,\,$ since they divide each other by above. In particular, they are equal if we are in $\Bbb Z$ and we normalize gcds to be positive. We used the GCD Universal Property (it often simplifies proofs, e.g. see this proof of the GCD * LCM law).

Or by this Remark it's case $\,m = abc\,$ in $\,(m/a,m/b) = m/{\rm lcm}(a,b)\,$ for $\,m\,$ being any common multiple of $a,b;\,$ or, expressed equivalently $\,\gcd(a',b') = {\rm lcm}(a,b)'\,$ by cofactor duality.

Alternatively, comparing powers of primes in unique factorizations, it reduces to the following $$\begin{eqnarray} \min(a+x,\,b+x) &\,=\,& \min(a,b) + x\\ \rm expt\ analog\ of\ \ \ \gcd(a \,* x,\,b \,* x)&=&\rm \gcd(a,b)\,*x\end{eqnarray}\qquad\qquad\ \ $$

The proof is precisely the same as the prior proof, replacing gcd by min, and divides by $\,\le,\,$ and using the universal property of min instead of that of gcd, i.e.

$$\begin{eqnarray} {\rm employing}\quad\ c\le a,b&\iff& c\le \min(a,b)\\ \rm the\ analog\ of\quad\ c\ \, |\, \ a,b&\iff&\rm c\ \,|\,\ \gcd(a,b) \end{eqnarray}$$

Then the above proof translates as below, $\ $ with $\,\ m(x,y) := {\rm min}(x,y)$

$c \le a,b \!\iff\!c\!+\!x \le a\!+\!x,b\!+\!x\!$ $\!\iff\! c\!+\!x \le m(a\!+\!x,b\!+\!x)\!$ $\!\iff\!\! c \le m(a\!+\!x,b\!+\!x)\!-\!x$

Or in any UFD we can induct on the number $\,k\,$ of prime factors of $\,x.\,$ If $\,k=0\,$ then $\,x\,$ is a unit thus $\,(ax,bx)=(a,b)x\,$ is clear. Else $\,x = yp\,$ for $\,p\,$ prime so by $\,\color{#c00}{I =\rm induction}$

$\begin{align} {\rm if}\,\ p\nmid d\,\ &{\rm then}\,\ d\mid ayp,byp \iff\,\ d\mid ay,by \color{#c00}{\overset{ I\!\!\!}\iff}\ d\mid (a,b)y \iff d\mid (a,b)yp\\ {\rm if}\,\ p\mid d\,\ &{\rm then}\,\ d\mid ayp,byp \iff \frac{d}p\mid ay,by \color{#c00}{\overset{ I\!\!\!}\iff} \frac{d}p\mid (a,b)y \iff d\mid (a,b)yp\end{align}$

Theorem $\ \ $ If $\ a,b,x\ $ are positive naturals then $\ (ax,bx) = (a,b)x $

Proof $\ $ We induct on $\color{#0a0}{{\rm size}:= a\!+b}.\,$ If $\,a=b\,$ then it is true since both sides $= ax.\,$ Else $\,a\neq b;\,$ wlog, by symmetry, $\,a > b\,$ so $\,(ax,bx) = (ax\!-\!bx,bx) = \color{}{((a\!-\!b)x,bx)}\,$ with smaller $\rm\color{#0a0}{size}$ $\,(a\!-\!b) + b = a < \color{#0a0}{a\!+b},\,$ therefore $\,((a\!-\!b)x,bx)\!\underset{\rm induct}=\! (a\!-\!b,b)x = (a,b)x$.

For completeness below we present a proof of the LCM Distributive Law with notation $\,[x,y] := {\rm lcm}(x,y),\,$ using the LCM universal property and basic divisibility properties.

Theorem $\ \ [ax,bx] \approx [a,b]x $

Proof $\ \ \smash{[ax,bx]\mid c\!\!\overset{\rm def\!\!\!}\iff ax,bx\mid c\iff a,b\mid c/x\!\!\overset{\rm def\!\!\!}\iff [a,b]\mid c/x\iff [a,b]x\mid c}$

Therefore $\rm\, [ax,bx] \approx [a,b]x\,$ are associate (divide each other) by above. In particular, they are equal if we are in $\Bbb Z$ and we normalize lcms to be positive.

It's simpler, you don't need any identity, factorization, almost nothing apart from the fact the $\gcd$'s exist.

Let $c=\gcd(a,b)$ and $d=\gcd(ma,mb)$, then $$c\mid a, b\Longrightarrow mc\mid ma, mb\Longrightarrow mc\mid d$$ so $d=mcx$, then $$mcx\mid ma, mb\Longrightarrow cx\mid a,b\Longrightarrow cx\mid c\Longrightarrow x\mid1$$

therefore $\gcd(ma, mb)=|m|\gcd(a,b)$.

Quite simply: Let $x=gcd(a,b).$ Then $a=xk$ and $b=xl$ for some coprime $k, l$ by definition of GCD.

Then $ma=mxk$ and $mb=mxl$.

Therefore, $gcd(ma,mb)=mx=mgcd(a,b)$ [because $gcd(k,l)=1$, so the greatest divisor of both $ma$, $mb$ must then be $mx$]. $\blacksquare$