HP 40gs User Manual page 286

hp40g+.book Page 12 Friday, December 9, 2005 1:03 AM
16-12
( )
GCD c , b = GCD c
n n
Part 2
Given the equation:
⋅ ⋅
b x + c y = 1
3 3
where the integers x and y are unknown and b
are defined as in part 1 above:
1. Show that [1] has at least one solution.
2. Apply Euclid's algorithm to b
solution to [1].
3. Find all solutions of [1].
Solution: Equation [1] must have at least one solution,
as it is actually a form of Bézout's Identity.
In effect, Bézout's Theorem states that if a and b are
relatively prime, there exists an x and y such that:
a x ⋅ b y ⋅
+ = 1
b
Therefore, the equation
one solution.
Now enter IEGCD(B(3),
C(3)).
Note that the IEGCD
function can be found on
the INTEGER submenu of
the MATH menu.
Pressing a number
of times returns the result
shown at the right:
In other words:
× × – (
b 1000 + c 999
3 3
Therefore, we have a particular solution:
x = 1000, y = –999.
The rest can be done on paper:
c = b + 2 b = 999 2
,
3 3 3
( ) ( )
2 , = GCD b 2 , = 1
n n
[1]
and c
3
and c and find a
3 3
x ⋅ y ⋅
+ c = 1
has at least
3 3
)
= 1
×
+ 1
Step-by-Step Examples
3