How many pairs (x,y) of positive integers satisfiy 2x+7y=1000

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- May 27th 2007, 09:46 PMDragonPairs of (x,y)
How many pairs (x,y) of positive integers satisfiy 2x+7y=1000

- May 27th 2007, 11:07 PMearboth
- May 28th 2007, 12:35 AMCaptainBlack

so for to be an integer must be a multiple of ,

but as this is even must be a multiple of .

But as ranges over the integers (which is the range

that is constrained to) ranges over .

Of the numbers exactly are divisible by 7,

so there are positive integer pairs that satisfy the

given equation.

RonL - May 28th 2007, 06:31 AMSoroban
Hello, Dragon!

My approach is a variation of CaptainBlack's . . .

Quote:

How many pairs (x,y) of positive integers satisfy

We have: .

Since is a positive integer, we have:.

There are: multiples-of-seven which are less than 1000

. . and__half__of them are even.

Therefore, there are**71**solutions.

- May 28th 2007, 08:11 AMThePerfectHacker

We can solve this using Bezout's Identity.

Trivially we see that,

Thus,

.

Hence,*all*integer solutions are given by,

.

We want positive solutions, meaning,

Since, is an integer, we need to find all integers so that,

We see that,

Are all these integers.

In total we have 71 positive solutions.