Question 563311
19a + 47b = 1000


Solving for a,


*[tex \LARGE a = \frac{1000 - 47b}{19}]


We want 1000-47b to be congruent to 0 mod 19. Since 1000 is 12 mod 19, 47b must be 12 mod 19. Also, 47 is congruent to 9 (mod 19), so we have


*[tex \LARGE 9b \equiv 12(mod 19)]


If b = 14, then 9b = 126 which is 12 mod 19, so this works. Hence 47b = 47*14 = 658, 19*18 = 342. If you know Diophantine equations you can use this solution to generate infinitely many solutions.