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Unformatted text preview: ( x-x ) and gcd ( a d , b d ) = 1, then b d | ( x-x ) by Prop. 2.28. Therefore, x-x = n b d for some n Z , so x = x + n b d . Substituting back into the equation above, we get a d n b d =-b d ( y-y ) or y = y-n a d . Therefore, x = x + n b d and y = y-n a d for some n Z , so every solution is of the required form. The last question that we have to ask is whether everything of this form is actually a so-lution. To determine this, we substitute. Choose any integer n . Then a x + n b d + b y-n a d = ax + by + an b d-bn a d = ax + by = d so each is actually a solution. Therefore, the complete solution is as we stated above....
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- Fall '08