Download Homework 5, due October 4

HOMEWORK #5
Problem 1. Find
GCD(a, b)
of each of each pair a and b and write it as an itegral linear combination of a and b.
(a) a = 25, b = 65
(b) a = 6166, b = 6105
Problem 2. For each pair a and b, determine if it is or is not possible to find integers u and
v so that
d = u × a + v × b.
(d as indicated).
(a) a = 25,b = 65,d = 115
(b) a = 2510 , b = 6512 ,d = 2 × 510
(c) a = 2510 , b = 6512 ,d = 2 × 511
Problem 3. Problem 24 of section 4.2.
When we run the Euclidean algorithm on a and b, common notation is as follows:
a = q1 b + r1 ,
set
a2 = b, b2 = r1 ;
set a3 = b2 , b3 = r2 ;
..
.
= qn−1 bn−1 + rn−1 , set an = bn−1 , bn = rn−1 ;
an = qn bn + 0;
a 2 = q 2 b2 + r 2 ,
an−1
and then
gcd(a, b)
=
=
···
=
=
=
gcd(a2 , b2 )
gcd(a3 , b3 )
gcd(an−1 , bn−1 )
gcd(an , bn )
bn
Let’s call
a3 , a4 , . . . , an
1
2
HOMEWORK #5
the intermediate values in the Euclidean algorithm. Notice that if n = 1, 2 there are no
intermediate values.
Problem 4. We call a natural number a perfect square if it is the square of another number.
(a) True or false: if a and b are perfect squares than the intermediate values in the
Eulclidean algormithm are all perfect squares.
(b) True or false: if a and b are perfect squares than their GCD is always a perfect
square.
Problem 5. Problem 9 in section 4.4, parts (b), (c), (d) and (i).