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, ?
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7?
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, CHAPTER 2 INTRODUCTION TO NUMBER
THEORY
tu
ANSWERS TO QUESTIONS
d
2.1 A nonzero b is a divisor of a if a = mb for some m, where a, b, and m
yg
are integers. That is, b is a divisor of a if there is no remainder on
division.
2.2 It means that b is a divisor of a.
en
2.3 In modular arithmetic, all arithmetic operations are performed modulo
some integer.
iu
2.4 An integer p > 1 is a prime number if and only if its only divisors are ±1
and ±p.
2.5 Euler's totient function, written f(n), is the number of positive integers
s2
less than n and relatively prime to n.
2.6 The algorithm takes a candidate integer n as input and returns the
result "composite" if n is definitely not a prime, and the result
47
"inconclusive" if n may or may not be a prime. If the algorithm is
repeatedly applied to a number and repeatedly returns inconclusive,
then the probability that the number is actually prime increases with
each inconclusive test. The probability required to accept a number as
??
prime can be set as close to 1.0 as desired by increasing the number of
tests made.
2.7 If r and n are relatively prime integers with n > 0. and if f(n) is the
??
least positive exponent m such that am º 1 mod n, then r is called a
primitive root modulo n.
2.8 The two terms are synonymous.
??
ANSWERS TO PROBLEMS
?
2.1 The equation is the same. For integer a < 0, a will either be an integer
multiple of n of fall between two consecutive multiples qn and (q + 1)n,
?
where q < 0. The remainder satisfies the condition 0 ≤ r ≤ n.
-8-
© 2020 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
, 2.2 In this diagram, q is a negative integer.
n
tu
(q+1)n a qn –3n –2n –n
–2 –1 0
r
d
2.3 a. 2 b. 3 c. 4 There are other correct answers.
yg
2.4 Section 2.3 defines the relationship: a = n ´ ëa/nû + (a mod n). Thus,
we can define the mod operator as: a mod n = a – n ´ ëa/nû.
a. 5 mod 3 = 5 – 3 ë5/3û = 2
en
b. 5 mod –3 = 5 – (–3) ë5/(–3)û = –1
c. –5 mod 3 = –5 – 3 ë(–5)/3û = 1
d. –5 mod –3 = –5 – (–3) ë(–5)/(–3)û = –2
2.5 a = b
iu
2.6 Recall Figure 2.1 and that any integer a can be written in the form
s2
a = qn + r
where q is some integer and r one of the numbers
47
0, 1, 2, . . ., n – 1
Using the second definition, no two of the remainders in the above list
are congruent (mod n), because the difference between them is less
??
than n and therefore n does not divide that difference. Therefore, two
numbers that are not congruent (mod n) must have different
remainders. So we conclude that n divides (a – b) if and only if a and b
are numbers that have the same remainder when divided by n.
??
2.7 1, 2, 4, 6, 16, 12
??
? ?
-9-
© 2020 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
??
7?
s24
niu
ge
, ?
??
7?
s24
niu
ge
, CHAPTER 2 INTRODUCTION TO NUMBER
THEORY
tu
ANSWERS TO QUESTIONS
d
2.1 A nonzero b is a divisor of a if a = mb for some m, where a, b, and m
yg
are integers. That is, b is a divisor of a if there is no remainder on
division.
2.2 It means that b is a divisor of a.
en
2.3 In modular arithmetic, all arithmetic operations are performed modulo
some integer.
iu
2.4 An integer p > 1 is a prime number if and only if its only divisors are ±1
and ±p.
2.5 Euler's totient function, written f(n), is the number of positive integers
s2
less than n and relatively prime to n.
2.6 The algorithm takes a candidate integer n as input and returns the
result "composite" if n is definitely not a prime, and the result
47
"inconclusive" if n may or may not be a prime. If the algorithm is
repeatedly applied to a number and repeatedly returns inconclusive,
then the probability that the number is actually prime increases with
each inconclusive test. The probability required to accept a number as
??
prime can be set as close to 1.0 as desired by increasing the number of
tests made.
2.7 If r and n are relatively prime integers with n > 0. and if f(n) is the
??
least positive exponent m such that am º 1 mod n, then r is called a
primitive root modulo n.
2.8 The two terms are synonymous.
??
ANSWERS TO PROBLEMS
?
2.1 The equation is the same. For integer a < 0, a will either be an integer
multiple of n of fall between two consecutive multiples qn and (q + 1)n,
?
where q < 0. The remainder satisfies the condition 0 ≤ r ≤ n.
-8-
© 2020 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
, 2.2 In this diagram, q is a negative integer.
n
tu
(q+1)n a qn –3n –2n –n
–2 –1 0
r
d
2.3 a. 2 b. 3 c. 4 There are other correct answers.
yg
2.4 Section 2.3 defines the relationship: a = n ´ ëa/nû + (a mod n). Thus,
we can define the mod operator as: a mod n = a – n ´ ëa/nû.
a. 5 mod 3 = 5 – 3 ë5/3û = 2
en
b. 5 mod –3 = 5 – (–3) ë5/(–3)û = –1
c. –5 mod 3 = –5 – 3 ë(–5)/3û = 1
d. –5 mod –3 = –5 – (–3) ë(–5)/(–3)û = –2
2.5 a = b
iu
2.6 Recall Figure 2.1 and that any integer a can be written in the form
s2
a = qn + r
where q is some integer and r one of the numbers
47
0, 1, 2, . . ., n – 1
Using the second definition, no two of the remainders in the above list
are congruent (mod n), because the difference between them is less
??
than n and therefore n does not divide that difference. Therefore, two
numbers that are not congruent (mod n) must have different
remainders. So we conclude that n divides (a – b) if and only if a and b
are numbers that have the same remainder when divided by n.
??
2.7 1, 2, 4, 6, 16, 12
??
? ?
-9-
© 2020 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist.
No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
