Math Help - Equation with FIVE VARIABLES!!

1. 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)

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

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?

2. 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.