Custom align environment which auto breaks line before = and aligns at =

The customalign is defined using newenviron in order to tap into the \BODY of the environment. Every = is replaced by \\ &= via \xpatchcmd*. A second \xpatchcmd then restores the first global replacement back to only &=:

\documentclass{article}

\usepackage{mathtools,regexpatch,environ}
\DeclareMathOperator{\lcm}{lcm}

\NewEnviron{customalign}{%
  \xpatchcmd*{\BODY}{=}{\\ &=}{}{}% Global replacement "=" > "\\ &="
  \xpatchcmd{\BODY}{\\ &=}{&=}{}{}% First replacement "\\ &=" > "&="
  \begin{align*}
    \BODY% Set align with replacement
  \end{align*}
}

\begin{document}

\begin{align*}
  \lcm(13, 7) &= (13 \cdot 7) : \gcd(13, 7) \\
              &= (13 \cdot 7) : 1 \\
              &= \overset{10 + 3}{13} \cdot 7 \\
              &= 10 \cdot 7 + 3 \cdot 7 \\
              &= 70 + 21 \\
              &= \boxed{91}
\end{align*}

\begin{customalign}
   \lcm(13, 7) = (13 \cdot 7) : \gcd(13, 7) 
               = (13 \cdot 7) : 1 
               = \overset{10 + 3}{13} \cdot 7 
               = 10 \cdot 7 + 3 \cdot 7 
               = 70 + 21 
               = \boxed{91}
\end{customalign}

\end{document}

This implementation avoids cumbersome replacements:

1. the body of the environment is split at =;
2. the first item is set apart and passed to split followed by &=;
3. the remaining item are delivered, separated by \\ &=.

\documentclass{article}
\usepackage{mathtools}
\usepackage{xparse}

\DeclareMathOperator{\lcm}{lcm}

\ExplSyntaxOn

\NewDocumentEnvironment{breakalign}{b}
 {
  \begin{equation}
  \soundwave_breakalign:nnn { #1 } { split } { }
  \end{equation}
 }
 {}
\NewDocumentEnvironment{breakalign*}{b}
 {
  \begin{equation*}
  \soundwave_breakalign:nnn { #1 } { split } { }
  \end{equation*}
 }
 {}
\NewDocumentEnvironment{breakaligned}{O{c} b}
 {
  \soundwave_breakalign:nnn { #2 } { aligned } { #1 }
 }
 {}

\tl_new:N \l__soundwave_breakalign_left_tl
\seq_new:N \l__soundwave_breakalign_body_seq

\cs_new_protected:Nn \soundwave_breakalign:nnn
 {
  \seq_set_split:Nnn \l__soundwave_breakalign_body_seq { = } { #1 }
  \seq_pop_left:NN \l__soundwave_breakalign_body_seq \l__soundwave_breakalign_left_tl
  \tl_if_empty:nTF { #3 } { \begin{#2} } { \begin{#2}[#3] }
  \l__soundwave_breakalign_left_tl &=
  \seq_use:Nn \l__soundwave_breakalign_body_seq { \\ &= }
  \end{#2}
 }

\ExplSyntaxOff

\begin{document}

\begin{breakalign*}
   \lcm(13, 7) = (13 \cdot 7):\gcd(13, 7)
               = (13 \cdot 7):1
               = \overset{10+3}{13} \cdot 7
               = 10 \cdot 7+3 \cdot 7
               = 70+21
               = \boxed{91}
\end{breakalign*}

\begin{breakalign}
   \lcm(13, 7) = (13 \cdot 7):\gcd(13, 7)
               = (13 \cdot 7):1
               = \overset{10+3}{13} \cdot 7
               = 10 \cdot 7+3 \cdot 7
               = 70+21
               = \boxed{91}
\end{breakalign}

\begin{align*}
&\begin{breakaligned}
   \lcm(13, 7) = (13 \cdot 7):\gcd(13, 7)
               = (13 \cdot 7):1
               = \overset{10+3}{13} \cdot 7
               = 10 \cdot 7+3 \cdot 7
               = 70+21
               = \boxed{91}
\end{breakaligned}
&&
\begin{breakaligned}
   \lcm(13, 7) = (13 \cdot 7):\gcd(13, 7)
               = (13 \cdot 7):1
               = \overset{10+3}{13} \cdot 7
               = 10 \cdot 7+3 \cdot 7
               = 70+21
               = \boxed{91}
\end{breakaligned}
\end{align*}

\end{document}

There are three environments:

1. breakalign will get a number;
2. breakalign* will be unnumbered;
3. breakaligned can be used as a subsidiary environment and accepts the usual [t] or [b] options like aligned for vertical alignment.

A version compatible with older (but not too old) versions of TeX Live.

\documentclass{article}
\usepackage{mathtools}
\usepackage{xparse}
\usepackage{environ}

\DeclareMathOperator{\lcm}{lcm}

\ExplSyntaxOn

\NewEnviron{breakalign}
 {
  \begin{equation}
  \soundwave_breakalign:Vnn \BODY { split } { }
  \end{equation}
 }

\NewEnviron{breakalign*}
 {
  \begin{equation*}
  \soundwave_breakalign:Vnn \BODY { split } { }
  \end{equation*}
 }

\NewEnviron{breakaligned}[1][c]
 {
  \soundwave_breakalign:Vnn \BODY { aligned } { #1 }
 }


\tl_new:N \l__soundwave_breakalign_left_tl
\seq_new:N \l__soundwave_breakalign_body_seq

\cs_new_protected:Nn \soundwave_breakalign:nnn
 {
  \seq_set_split:Nnn \l__soundwave_breakalign_body_seq { = } { #1 }
  \seq_pop_left:NN \l__soundwave_breakalign_body_seq \l__soundwave_breakalign_left_tl
  \tl_if_empty:nTF { #3 } { \begin{#2} } { \begin{#2}[#3] }
  \l__soundwave_breakalign_left_tl &=
  \seq_use:Nn \l__soundwave_breakalign_body_seq { \\ &= }
  \end{#2}
 }
\cs_generate_variant:Nn \soundwave_breakalign:nnn { V }

\ExplSyntaxOff

\begin{document}

\begin{breakalign*}
   \lcm(13, 7) = (13 \cdot 7):\gcd(13, 7)
               = (13 \cdot 7):1
               = \overset{10+3}{13} \cdot 7
               = 10 \cdot 7+3 \cdot 7
               = 70+21
               = \boxed{91}
\end{breakalign*}

\begin{breakalign}
   \lcm(13, 7) = (13 \cdot 7):\gcd(13, 7)
               = (13 \cdot 7):1
               = \overset{10+3}{13} \cdot 7
               = 10 \cdot 7+3 \cdot 7
               = 70+21
               = \boxed{91}
\end{breakalign}

\begin{align*}
&\begin{breakaligned}
   \lcm(13, 7) = (13 \cdot 7):\gcd(13, 7)
               = (13 \cdot 7):1
               = \overset{10+3}{13} \cdot 7
               = 10 \cdot 7+3 \cdot 7
               = 70+21
               = \boxed{91}
\end{breakaligned}
&&
\begin{breakaligned}
   \lcm(13, 7) = (13 \cdot 7):\gcd(13, 7)
               = (13 \cdot 7):1
               = \overset{10+3}{13} \cdot 7
               = 10 \cdot 7+3 \cdot 7
               = 70+21
               = \boxed{91}
\end{breakaligned}
\end{align*}

\end{document}