HP 40gs, User Manual

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16-12 Step-by-Step Examples
Part 2
Given the equation:
[1]
where the integers x and y are unknown and b
3
and c
3
are defined as in part 1 above:
1. Show that [1] has at least one solution.
2. Apply Euclid’s algorithm to b
3
and c
3
and find a
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:
Therefore, the equation has at least
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:
Therefore, we have a particular solution:
x
= 1000,
y
= –999
.
The rest can be done on paper:
,
GCD c
n
b
n
,
( )
GCD c
n
2 ,
( )
GCD b
n
2 ,
( )
1 = = =
b
3
x c
3
y 1 =
⋅
+
⋅
a x
⋅
b y
⋅
+ 1 =
b
3
x
⋅
c
3
y
⋅
+ 1 =
b
3
1000
×
c
3
999 –
( ) ×
+ 1 =
c
3
b
3
= 2 + b
3
999 2 1 +
×
=
hp40g+.book Page 12 Friday, December 9, 2005 1:03 AM