a) $ 130x + 143y = 5957 $
b) $ 44x + 19y = 75 $
the theorem says ax + by = c has solution if and only if d | c
however I work out both question with no solution as
a)
$ 143 = 130 . 1 + 13 $
$ 130 = 13 . 10 + 0 $
and 5957 is unable to be divided by 13 therefore no solution exists.
b)
$ 44 = 19 . 2 + 6 $
$ 19 = 6 . 3 + 1 $
$ 6 = 3 . 2 + 0 $
for b) I am unsure if there are any solution exists?