Unformatted text preview: 0. Since ab < 0, either a > , b < 0 or a < , b > 0. Recall that d > 0. Suppose a < , b > 0. Thena d > , b d > 0 and hence x = n b d + x > 0 and y = n (a d ) + y > 0 if n >x b d and n >ya d = y a d . This is true for all n > maximum(x b d , y a d ) giving inﬁnite number of positive solutions. Suppose a > , b < 0. Then a d > ,b d > 0 and hence x = (n )(b d )+ x > 0 and y = (n )( a d )+ y > 0 ifn >xb d = x b d andn >y a d . or if n <x b d and n <y a d . This is true for all n < minimum(x b d ,y a d ) giving inﬁnite number of positive solutions. Therefore whenever a, b ∈ Z , ab < 0 and gcd( a, b )  c , we get inﬁnite number of positive solutions. ± Remark: Whenever ab > , we cannot get an inﬁnite number of positive solutions for any c ....
View
Full
Document
This note was uploaded on 11/25/2010 for the course MATH 135 taught by Professor Andrewchilds during the Winter '08 term at Waterloo.
 Winter '08
 ANDREWCHILDS
 Math

Click to edit the document details