Find all solutions of 123x 57y 531Solution123x57y 531 123

Find all solutions of 123x + 57y = 531.

123x+57y = 531

123=41x3

and 57 = 19x3

Hence we can rewrite the equation as

41x+19y = 177

By trial and error we find that when x =2

41(2)+19y = 177

Or 19y = 95

y =5

Hence (2,5) = (x,y) satisfies this

coefficient of y is prime = 19

if we increase of decrease x by 19 or multiples of 19 we get integral answers for y

x=21, y = -36, x=40, y =-77, x = -19, y = -17, ,,,,

Or solutions would be of the form

x = 2+19z where z belongs to set of integers

and y =5-41z