In: Advanced Math
3. Let a and b be positive integers. Is gcd(5a + b, 11a + 2b) = gcd(2a + b, 3a + 2b)?
If yes provide a proof. If not, provide a counterexample.
Solution:
Let
be an integer matrix and let
Then, we have
(Cramer's Rule)
where
is the adjoint matrix (the transpose of the cofactor matrix), and
so
(From
and the
Associativity of Matrix multiplication)
Let
and
. Then
by the first matrix equation
( ,
then
)
and
by the second matrix equation
.
In particular,
if
.
Here, we have
(Since
)
Similarly, we have
(Since
)
Hence