§
2.4 Problem 25 (
a
):
φ
(4) = 2.
(
b
):
φ
(10) = 4.
(
c
):
φ
(13) = 12.
§
2.4 Problem 26: If
n
is prime then it is relatively prime to all the numbers 1
,
2
,
3
, . . .,
(
n
−
1), so
φ
(
n
) =
n
−
1. If
n
= 1 then
φ
(1) = 1
negationslash
= 1
−
1. If
n
is not prime and
n >
1 then it has some factor
a < n
, and gcd(
a, n
) =
a
negationslash
= 1.
Thus one of the numbers 1
,
2
,
3
, . . .,
(
n
−
1) is not relatively prime to
n
, and
φ
(
n
)
negationslash
=
n
−
1.
§
2.4 Problem 27:
If
p
is prime then the only prime factor of
p
k
is
p
.
Thus the only numbers that are not
relatively prime to
p
k
are the numbers that have
p
as a factor. The numbers less than or equal to
p
k
that
are not relatively prime to
p
k
are the numbers
p,
2
p,
3
p,
4
p, . . ., p
k
,
and there are
p
k

1
of these. Thus
φ
(
p
k
) =
p
k
−
p
k

1
.
§
2.4 Problem 30 (
a
): lcm(2
2
·
3
3
·
5
5
,
2
5
·
3
3
·
5
2
) = 2
5
·
3
3
·
5
5
.
(
b
): lcm(2
·
3
·
5
·
7
·
11
·
13
,
2
11
·
3
9
·
11
·
17
14
) = 2
11
·
3
9
·
5
·
7
·
11
·
13
·
17
14
.
(
c
): lcm(17
,
17
17
) = 17
17
.
(
d
): lcm(2
2
·
7
,
5
3
·
13) = 2
2
·
5
3
·
7
·
13.
(
e
): lcm(0
,
5) is undefined. (It can’t be 0, because the LCM is defined as a positive integer.) However, some
texts would define it to be 0 in this case.
(
f
): lcm(2
·
3
·
5
·
7
,
2
·
3
·
5
·
7) = 2
·
3
·
5
·
7.
§
2.5 Problem 19: First we need to find the binary representation of 2003.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '06
 CALLAHAN
 Harshad number, Prime number, Euclidean algorithm

Click to edit the document details