Equation with FIVE VARIABLES!!

Hi. I've been practicing solving equations these days and today, I found this problem. Here it is.


Solve this equation. (a, b, c, d, e are all natural numbers)

$\displaystyle abcde-a=357^{400}$
$\displaystyle abcde-b=359^{410}$
$\displaystyle abcde-c=361^{420}$
$\displaystyle abcde-d=363^{430}$
$\displaystyle abcde-e=365^{440}$


Well, I can solve an equation with two or three variables but
what should I do if the number of variables are more than three?

I would expect that any of the normal elimination or substitution methods can be applied. However, I would expect that whatever the method is involving matrices will be much faster.