Let $a,b,c,d$ be positive integers such that$abcd=8!$. If
$$
\begin{align*}
ab+a+b&=524\
bc+b+c&=146\
cd+c+d&=104
\end{align*}
$$
Find $a-d$.

Question

Let $a,b,c,d$ be positive integers such that$abcd=8!$. If 
$$
\begin{align*}
ab+a+b&=524\
bc+b+c&=146\
cd+c+d&=104
\end{align*}
$$
Find $a-d$.

watchmath · Answer

Let $a,b,c,d$ are positive integers such that $abcd=8!$. If
$$
\begin{align*}
ab+a+b&=524\
bc+b+c&=146\
cd+c+d&=104
\end{align*}
$$
Find $a-d$.

mathteacher1729 · Answer

You can use \align environment in here? wow. :)

watchmath · Answer

yes :D

mathteacher1729 · Answer

That is $\mathbb{R}\text{eally}$ cool.

mathteacher1729 · Answer

Why do I feel like there is a more elegant way of solving this than using Lagrange multipliers...

watchmath · Answer

This is an AMC 12 problem so no need for Lagrange. I am not sure that you can use Lagrange to solve this since we are not looking extreme value.

mathteacher1729 · Answer

I saw a system of nonlinear equations with what appears to be a constraint $abcd=8!$ and thought "lagrange".  :(

mathteacher1729 · Answer

I recognize that $8! = 2^7×3^2×5×7$