. Suppose that c and d are given integers with d 0. There are integers x and y such that (x,y) = d...

Question

. Suppose that c and d are given integers with d > 0. There are integers x and y such that (x,y) = d and xy = c if and only if d2 c. Let d = (a, b). Then a I be if and only if -Qd- is a divisor of c.
Problem 3. Consider a function f : D --* 111, and suppose that for every a E D there exist Ea > 0 and a neighborhood Na of a such that f(x) > Ea for all x E Na, n D. . Prove that if D is a compact, then,there exists E > 0 such that f (x) > E for all x E D. Give an example to show that :the conclusion of the previous statement need not hold if the assumption that D is compact is omitted.

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Robert · Accepted Answer

The answer is FALSE, that is for 
,
cd
Î
¢
 there do not exist integers 
,
xy
 such that
(
)
,
xyd
=
 and 
xyc
=
 if and only if 
2
|
dc
Suppose it is true that for 
,
cd
Î
¢
 there exist integers 
,
xy
 such that 
(
)
,

. Suppose that c and d are given integers with d > 0. There are integers x and y such that (x,y) = d and xy = c if and only if d2 c. Let d = (a, b). Then a I be if and only if -Qd- is a divisor of c....

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