Section 34
1.
For the pairs of integer
a
,
b
q
and
r
such
that
a
=
qb
+
r
and
0
r < b
(a)
a
= 100
and
b
= 3
.
q
= 33
and
r
= 1
.
(b)
a
=
±
100
and
b
= 3
.
q
=
±
34
and
r
= 2
.
(c)
a
= 99
and
b
= 3
.
q
= 33
and
r
= 0
.
(d)
a
=
±
99
and
b
= 3
.
q
=
±
33
and
r
= 0
.
(e)
a
= 0
and
b
= 3
.
q
= 0
and
r
= 0
.
There is only one correct answer to each of these.
2.
For each of the pairs of integers
a
,
b
in the previous problem, compute
a
div
b
and
a
mod
b
.
This is the same question, except you have to know what
a
div
b
and
a
mod
b
are. In fact,
a
div
b
is the
q
a
mod
b
is the
r
. So, for example, the answer to (b) is that
±
100
div
3 =
±
34
and
±
100 mod 3 = 2
.
Section 35
1. Please calculate:
(a)
gcd (20
;
25)
When the two numbers are positive integers, I like to arrange this
computation as follows. Start out with the two numbers
20
and
25
in descending order:
25
,
20
Then compute
25 mod 20 = 5
to get
25
,
20
,
5
1
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View Full DocumentThen compute
20 mod 5 = 0
. That tells you that
5 = gcd (20
;
25)
.
At each step you compute
x
mod
y
where
x
and
y
are the last two
numbers in your list. You stop when
x
mod
y
= 0
, at which point
y
is the desired greatest common divisor.
(b)
gcd (0
;
10)
When one of the numbers is
0
use the fact that
gcd (0
; b
) =
b
for
any integer
b
. So
gcd (0
;
10) = 10
.
(c)
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 Spring '08
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 Prime number, Greatest common divisor, gcd

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