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?
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
The obervation of Plato is correct, so that i have first to apologize ... 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