solve the diophantine equation, 83x+111y=4626

Question

the question is obtained from number theory

Answer 1

x=33, y=17.

Find integers x and y satisfying (61-28y)/83 is an integer (positive or negative).

When 83 is divided into 4626, the remainder is 61; when 83 is divided into 111, the remainder is 28, so the original equation is reduced by considering only the remainders. 

Then x=55+(61-28y)/83-y.

When y=17, 61-28y=-415=-5×83, so x=55-5-17=33.