Prove (15n+17,10n+11) = 1

Question

I am new to number theory and wanted to know If I am doing this correctly:

W.t.s. $(15n+17,10n+11) = 1$

Using the Division Algorithm we have

$15n + 17 = (10n + 11)(1) + (5n + 6)$

$10n + 11 = (5n + 6)(1) + (5n + 5)$

$5n + 6 = (5n + 5) + 1$

$\Rightarrow$ $(15n + 17,10n + 11)$ = ($5n + 5,1)$, but $(5n + 5,1)$ = $1$ (1 is relatively prime to every integer)

$\therefore$ $(15n+17,10n+11) = 1$ as needed?

Answer 1

Eliminate $n$

$$2(15n+17)-3(10n+11)=?$$