MTHT
467
MIDTERM‐‐ANSWERS
Spring
2011
NAME:
1.
12
points
True
or
False
Indicate
whether
the
statement
is
True
or
False.
If
False,
provide
reasoning
or
a
counterexample.
If
True,
briefly
explain
why.
i. T
or
F
If
x
and
y
are
integers
and
3|x
and
3|y,
then
3|(x
+y).
True.
Since
3
is
a
multiple
of
x
and
y,
it
is
also
(by
the
distributive
rule)
a
multiple
of
x+y.
ii. T
or
F
If
a,
b
and
c
are
positive
integers
and
a|b,
then
a|bc.
True.
If
a
is
a
factor
of
b,
it
is
also
a
factor
of
anything
times
b.
iii. T
or
F
(a + b) 2 = a 2 + b 2 Let a = 2 and b = 3, then
False.
It
suffices
to
give
a
counterexample:
(a + b)2 = (2 + 3)2 = 5 2 = 25, but a 2 + b 2 = 2 2 + 32 = 4 + 9 = 13
iv. T
or
F
If
a
>
b
and
gcd(a,b)=7,
then
gcd(b,
a‐b)=7.
Any
common
factor
of
both
a
and
b
must
also
be
a
factor
of
both
b
and
a‐b
(see
i)
and
vice
versa.
Since
the
two
sets
of
numbers
have
the
same
common
factors,
they
of
course
have
the
same
greatest
common
factor.
2.
8
points
Show
how
to
use
the
Euclidean
Algorithm
to
find
the
greatest
common
divisor
of
2321
and
847.
As
you
go,
explain
why
this
works
to
find
the
gcd.
NOTE:
the
arrow
part
is
not
needed.
We
apply
part
iv.
From
the
first
page
over
and
over
again
to
reduce
the
size
of
the
numbers
that
we
are
computing
the
gcd
for:
Computation:
Justification:
2321 − 2 ⋅ 847 = 627
847 − 1⋅ 627 = 220
627 − 2 ⋅ 220 = 187
220 − 1⋅187 = 33
187 − 5 ⋅ 33 = 22
33 − 1⋅ 22 = 11 ←
22 − 2 ⋅11 = 0
gcd(2321, 847) = gcd(847, 2321 − 2 ⋅ 847) = gcd(847, 627)
gcd(847, 627) = gcd(627, 220)
gcd(627, 220) = gcd(220,187)
gcd(220,187) = gcd(187, 33)
gcd(187, 33) = gcd(33, 22)
gcd(33, 22) = gcd(22,11)
gcd(22,11) = gcd(11, 0) = 11
Once
you
get
to
=0,
the
gcd
is
the
previous
number
computed,
in
this
case
11.
3.
8
points
Show
how
to
use
factor
trees
to
find
the
gcd
of
84
and
198.
84
198
4
21
2
99
2
2
3
7
9
11
3
3
Circle
all
common
prime
factors.
The
gcd
is
the
product
of
all
the
common
prime
factors
in
the
two
trees,
in
this
case
gcd(84,198)=2⋅3=6.
4.
8
points
Find
a
solution
to
the
equation:
94N+103M=1
Use
Euclidean
algorithm
or
the
calculator
program.
Answer
is
N=‐23,
M=21
Good idea to check: 94⋅
(‐23)+103⋅21
=
‐2146
+
2163
=
1
5.
8
points
Show
how
to
use
the
Sieve
of
Eratosthenes
to
find
all
the
prime
numbers
less
than
100.
List
your
steps
at
the
side.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
Cross off 0 and 1. They are
not prime numbers
Circle 2 – it is a prime number
because itʼs only factors are 1
and 2. Cross off all
multiples of 2.
Circle the next number that
is not crossed off (3) -- this
number is a prime number
because it would have been
crossed off if it has a factor
smaller than itself. Cross off
all multiples of that number.
Repeat the previous step
until the number you circle has
no multiples on the page left to
cross off. In this case it is the
number 11 – there are no
multiples of 11 left because 11⋅
11 is greater than 99 and all
smaller multiples of 11 have
already been crossed off.
Circle all the rest of the numbers that havenʼt been crossed off yet. They are all the prime
numbers less than 99 because they are not a multiple of any smaller number.
6.
8
points
You
only
know
the
three
given
numbers
in
a
combination
chart.
What
number
belongs
in
the
box
with
the
question
mark?
Why?
115
56
99
40
There
are
many
ways
to
see
this.
All
use
the
fact
that
an
arrow
cricket
makes
an
arithmetic
sequence,
always
adding
the
same
amount
no
matter
where
it
starts.
Consider
the
→→↑
cricket.
From
40
to
99
it
adds
59,
so
it
also
adds
59
from
56
to
?.
56+59=115.
7.
8
points
Model
the
following
arithmetic
statement
using
crickets
on
a
number
line:
Be
sure
to
indicate
whether
the
negative
sign
is
being
modeled
by
a
negative
cricket
or
by
moving
backwards.
Do
not
first
modify
the
arithmetic
statement,
but
show
how
to
the
statement
as
written.
3
–
(‐2)
=
5
The
first
negative
sign
means
go
backwards,
the
negative
sign
on
the
2
means
use
a
negative
cricket.
The
negative
cricket,
which
wants
to
go
to
the
left
is
directed
backwards
or
to
the
right
–
landing
on
5.
8.
8
points
Don
loves
peanut
butter
and
jelly
sandwiches.
One
day
while
he
was
eating,
he
noticed
that
each
jumbo
jar
of
peanut
butter
has
72
servings,
but
the
jelly
jar
has
only
40
servings.
If
he
opened
jars
on
the
same
day
and
used
exactly
one
serving
each
day,
how
many
days
would
it
take
until
he
emptied
a
peanut
butter
jar
and
a
jelly
jar
on
the
same
day?
The
answer
is
lcm(72,40)
=
360
9.
8
points
Today
is
a
Wednesday.
What
day
of
the
week
will
it
be
100
days
from
now?
100
=
7⋅14+2,
2
days
from
Wednesday
is
Friday
10.
8
points
A
customer
wants
to
mail
a
package.
The
postal
clerk
determines
the
cost
of
the
package
to
be
$1.02,
but
only
8¢
and
12¢
stamps
are
available.
Can
the
available
stamps
be
used
for
the
exact
amount
of
the
postage?
Why
or
why
not?
The
equation
8N+12M=”anything”,
has
a
solution
only
if
“anything”
is
a
multiple
of
the
gcd(8,12).
Since
gcd(8,12)=4
and
since
102
is
NOT
multiple
of
4,there
is
no
solution.
PLEASE
NOTE:
It
is
not
enough
to
say
that
102
is
not
a
multiple
of
either
8
or
12.
For
example,
one
can
make
52¢
by
using
5
8¢‐stamps
and
1
12¢‐stamp,
but
52
is
NOT
a
multiple
of
either
8
or
12.
11.
8
points
Terry
has
some
4
oz.
weights,
some
7
oz.
weights
and
a
two‐pan
balance.
Show
how
she
can
weigh
out
1
oz.
of
chocolate.
Put
2
4oz
weights
on
one
side
of
a
scale
and
1
7oz
weights
on
the
other.
Put
chocolate
on
the
side
with
the
7oz
weight
until
the
scales
balance.