Let a, b be relatively prime positive integers and consider the equation ax+by = ab.

(a) Show that the equation has no solution (x, y) belongs to N × N.

(b) Does it have a solution with (x, y) belongs to Z × Z? Why/why not?

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- April 26th 2009, 10:54 PMsmithhallIntegers
Let a, b be relatively prime positive integers and consider the equation ax+by = ab.

(a) Show that the equation has no solution (x, y) belongs to N × N.

(b) Does it have a solution with (x, y) belongs to Z × Z? Why/why not? - April 27th 2009, 12:25 AMchisigma
Let's consider the equation...

(1)

where and are relatively prime integers. Deviding both terms of (1) by b [the same is if we devide by a...] we obtain the equation...

(2)

Since and , no solution of (2) exist. Because , the same holds for ...

Kind regards

- April 27th 2009, 04:12 AMPlato
,

I am not sure what part of the question you thought you proved.

However, consider has many solutions in but no solution in . - April 27th 2009, 05:20 AMchisigma
The obervation of Plato is correct, so that i have first to apologize (Headbang)... and after to give the right solution...

Let's consider the equation...

... where and are relatively prime integers. Deviding both terms of (1) by b [the same is if we devide by a...] we obtain the equation...

(2)

Now if a and b are relatively prime in order to have in necessary that with k integer, i.e. x devides b. In order to have also and however, must be and that is a contadiction. The conclusion is that no solution exist and that infinite solutions exist under the condition with ...

Kind regards