Is there a positive integer solution to 1234x-4321y=1?

[scottstapp]

Homework Statement

Find a positive integer solution to 1234x-4321y=1, both x and y will be positive.

Homework Equations

The Attempt at a Solution

I created this array

4321 1234 619 615 4 3 1
3 1 1 153 1
1082 309 155 154 1 1 0

When plugging these (positive) values in I never get 1 I only get -1 when using x=1082 and y=309. Does this mean that no positive solution exists?

Thanks

 

[HallsofIvy]

You may just have an arithmetic error. I get x= 1182, not 1082.

 

[scottstapp]

Can you explain to me how you got 1182? Is my entire bottom row incorrect?

 

[HallsofIvy]

Oh, how embarassing! Your 1082 is completely correct. Apparently I made a silly arithmetic error myself.

You are correct, then, that 1082(1234)- 309(4321)= -1.

Multiplying through by -1 gives (-1082)(1234)- (-309)(4321)= 1.

But x= -1082 and y= -309 is not the only solution. If we were to add any multiple of 4321 to x and add the same multiple of 1234 to y, so that we have x+ 4321k and y- 1234k, then 1234(x+ 4321k)- 4321(y+ 1234k)= 1234x- 4321y+ ((1234)(4321)k- (4321)(1234)k)= 1234x- 4321y.

So just find k such that -1082+ 4321k and -309+ 1234k are positive. There are plenty of such solutions. Can you find the smallest?

 

[scottstapp]

In this problem can I actually just multiply through by -1 though? I am supposed to have a positive x and a positive y. So doesn't that mean that there does not exist any positive x and y such that 1234x-4321y=1? I know this seems to be a very elementary question but by the terms of this problem I am not sure if that is a "legal" move.

 

[HallsofIvy]

Multiplying by -1 gives negative solutions but my point was that you can then add any multiple of 4321 to the x value and 1234 to the y value and make the solutions positive.