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If x and y are positive integers, find the number of solution of the equation 3x + 5y = 1008

 Jun 15, 2020
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Lemma: If (x, y) is a solution to the equation 3x + 5y = 1008, where x and y are positive integers, then (x + 5, y - 3) is also a solution.

Proof: 3(x + 5) + 5(y - 3) = 3x + 15 + 5y - 15 = 3x + 5y = 1008.

 

By trial and error, we know that (1, 201) is a solution.

By Lemma, (6, 198), (11, 195), (16, 192), etc. are all solutions.

 

It is easy to know that (331, 3) is a solution.

 

The problem simplifies to "Finding the number of multiples of 3 among (3, 6, 9, 12, ..., 201)".

 

The rest is trivial.

 

Ans: 67

 Jun 15, 2020

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