SOLUTION: Your task is to find two different numbers that sum to 1000 . One number must be a multiple of 19 and the number must be a multiple of 47 . Show or explain how you got your answer

Question 563311: Your task is to find two different numbers that sum to 1000 . One number must be a multiple of 19 and the number must be a multiple of 47 . Show or explain how you got your answer .
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
19a + 47b = 1000

Solving for a,

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

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.
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