Problem:
Find the least common multiple of $\{41,-32\}$.
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Find the least common multiple}: \\
 \text{lcm}(41,-32) \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{Any }\text{multiple }\text{of }\text{a }\text{negative }\text{number }\text{is }\text{also }\text{a }\text{multiple }\text{of }\text{its }\text{absolute }\text{value}: \\
 \text{lcm}(41,-32)=\text{lcm}(41,32) \\
\end{array}
 \\
 
\begin{array}{l}
 
\begin{array}{l}
 \text{Find }\text{the }\text{prime }\text{factorization }\text{of }\text{each }\text{integer}: \\
 \text{The }\text{prime }\text{factorization }\text{of }\text{41 }\text{is}: \\
\end{array}
 \\
 41=41^1 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{The }\text{prime }\text{factorization }\text{of }\text{32 }\text{is}: \\
 32=2^5 \\
\end{array}
 \\
 
\begin{array}{l}
 
\begin{array}{l}
 \text{Find the largest power of each prime factor} \\
 \text{The }\text{largest }\text{power }\text{of }2 \text{that }\text{appears }\text{in }\text{the }\text{prime }\text{factorizations }\text{is }2^5 \\
 \text{The }\text{largest }\text{power }\text{of }\text{41 }\text{that }\text{appears }\text{in }\text{the }\text{prime }\text{factorizations }\text{is }41^1 \\
 \text{Therefore }\text{lcm}(41,32)=\left(2^5\times 41^1=1312\right): \\
\end{array}
 \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & \text{lcm}(41,-32)=1312 \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/number_theory/lcm_w_steps/756.txt'}