CHAPTER 3: ELEMENTARY NUMBER THEORY

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CHAPTER 3 ELEMENTARY NUMBER THEORY

• Fundamental Discrete Structure
• DCT 1073

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CONTENT

• 3.1 Divisibility
• 3.2 Primes
• 3.3 The Division Algorithm
• 3.4 Greatest Common Divisors (GCD)
• 3.5 Least Common Multiples (LCM)
• 3.6 Euclidean Algorithm
• 3.7 Modular Arithmetic
• 3.8 The Chinese Remainder Theorem

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OBJECTIVES

• At the end of this chapter you should be able to
• 3.1 Determine the divisibility of integers
• 3.2 Determine the prime factorization of an
  integer
• 3.3 Find the quotient and remainder from a
  division of integers
• 3.4 Find the greatest common divisor of two
  integers
• 3.5 Find the least common multiple of two
  integers
• 3.6 Use the Euclidean Algorithm to find the
  Greatest Common Divisor of two integers
• 3.7 Apply the use of congruences and modular
  arithmetic in hashing function, pseudorandom
  numbers and cryptology.
• 3.8 Find the smallest nonnegative integer that
  satisfies the given system of congruences.

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INTRODUCTION

• Number Theory is the branch of pure mathematics
  involving integers and their properties.
• The basic concepts of number theory discussed in
  this chapter are used widely throughout computer
  science.

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3.1 DIVISIBILITY
Lesson outcome Determine the divisibility of
integers
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Division in the integers

• When one integer is divided by a second, nonzero
  integer, the quotient may or may not be an
  integer.
• For example

Not an integer
Integer
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Division in the integers

• Definition

If a and b are integers with a ? 0 , we say that
a divides b if there is an integer c such that b
a c. When a divides b we say that a is a
factor of b and that b is a multiple of a.
The notation a b denotes that a divides
b. The notation denotes that a
does not divide b.
Remark We can express a b using quantifiers
as
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Example

• Determine whether 3 7 and 3 12
• 3 7 FALSE since 7/3 is not an integer
• 3 12 TRUE since 12/3 is an integer

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Theorem

• Let a, b and c be integers. Then,
• if a b and a c , then a (b c)
• if a b , then a bc for all integers c
• if a b and b c , then a c .

• For example

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Exercise 3.1

• Determine whether
• 7 21
• 4 15
• Does 13 divide each of these number?
• 79
• 113
• 5954
• Show that if a is an integer other than 0, then
• 1 divides a
• a divides 0

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3.2 PRIMES
Lesson outcome Determine the prime factorization
of an integer
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Prime

• Every positive integer greater than 1 is
  divisible by at least two integers, since a
  positive integer is divisible by 1 and by itself.
  
• Integers that have exactly two different positive
  integer factors are called primes.
• DEFINITION
• A positive integer p greater than 1 is
  called prime if the only positive factors of p
  are 1 and p. (EXAMPLE 5 IS PRIME)
• A positive integer that is greater than 1
  and is not prime is called composite. Integer n
  is composite iff there exists an integer a such
  that a n and 1 lt a lt n. (EXAMPLE 12 IS
  COMPOSITE)

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Primes
Example The integer 7 is prime since its only
positive factors are 1 and 7, whereas the
integer 9 is composite since it is divisible by 3.
The primes less than 100 are 2, 3, 5, 7, 11,
13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,
61, 67, 71, 73, 79, 83, 89, and 97.
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Prime Factorization

• Theorem 1The Fundamental Theorem of Arithmetic
• Every positive integer greater than 1 can be
  written uniquely as the product of primes.
• Theorem 2
• If n is a composite integer then n has a prime
  divisor less than or equal to .
• Theorem 3
• There are infinitely many primes

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Prime Factorization

• The prime factorization of 100, 641, 999, 1024
  and 7007 is given by

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Exercise 3.2

• Determine whether each of these integers is
  prime.
• 119 c) 101
• 277 d) 1113
• Find the prime factorization of these integers?
• 39 d) 238
• 113 e) 96
• 5954 f) 101

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3.3 THE DIVISION ALGORITHM
Lesson outcome Find the quotient and remainder
from a division of integers
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The Division Algorithm
a dividend, d divisor, q quotient, r
remainder

• q a div d r a mod d

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Examples

• What are the quotient and remainder when 101 is
  divided by 11?
• solution 101 11 9 2,
• the quotient is 9 101 div 1 and
• the remainder is 2 101 mod 11
• What are the quotient and remainder when -11 is
  divided by 3?
• solution -11 3(-4) 1,
• the quotient is -4 -11 div 3 and
• the remainder is 1 -11 mod 3
• Find the quotient and remainder when a 17 and d
  3
• solution so

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Exercise 3.3

• Find the quotient and remainder when
• a 4 and d 10
• a -13 and d 5
• What are the quotient and remainder when
• 44 is divided by 8?
• 777 is divided by 21?
• -2002 is divided by 87?

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3.4 GREATEST COMMON DIVISORS (GCD)
Lesson outcome Find the greatest common divisor
of two integers
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Greatest Common Divisor (GCD)

• DEFINITION
• If a, b, and k are in , and k a and k b
  , we say that k is a common divisor of a and b.
• If d is the largest such k, d is called the
  greatest common divisor, or GCD, of a and b, and
  we write d gcd (a, b).

TIPS
The largest integer that divides both of two
integers is called GREATEST COMMON DIVISOR of
these integers
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Example

• Find GCD(105,30).
• Solution
• The divisors of 30 are 1, 2, 3, 5, 6, 10, 15, and
  30.
• The divisors of 105 are 1, 3, 5, 7, 15, 21, 35,
  and 105.
• Therefore the common divisors of 30 and 105 are
  1, 3, 5, and 15.
• Hence, GCD(105,30) 15.

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Example

• What is the greatest common divisor of 24 and 36?
• Solution
• The divisor of 24 are 1, 2, 3, 4, 6, 8, 12, 24
• The divisor of 36 are 1, 2, 3, 4, 6, 9, 12, 18,
  36
• Therefore the common divisor of 24 and 36 are 1,
  2, 3, 4, 6, and 12.
• Hence, GCD (24, 36) 12.

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Exercise 3.4

• Find GCD(60,100).
• What is the greatest common divisor of 34 and 58?
• Find GCD(45,33).
• What is the greatest common divisor of 77 and
  128?

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Relatively Prime

• If GCD (a, b) 1, we say that a and b are
  relatively prime.

• EXAMPLE
• What is the greatest common divisor of 17 and
  22?
• Solution
• The divisor of 17 are 1, 17
• The divisor of 22 are 1, 2, 11, 22
• Therefore the common divisor of 17 and 22 is 1.
• Hence, GCD (17, 22) 1.

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Pairwise Relatively Prime

• The integers a1,a2,,an are pairwise relatively
  prime
• if GCD (ai, bj) 1, whenever 1 i lt j n.

• EXAMPLE
• Determine whether the integers 10, 17, and 21
  are pairwise relatively prime and whether the
  integers 10, 19, and 24 are pairwise relatively
  prime.
• Solution
• Since GCD(10, 17) 1, GCD(10, 21) 1, and GCD
  (17, 21) 1, we conclude that 10, 17, and 21 are
  pairwise relatively prime.
• Since GCD (10, 24) 2 gt 1, we see that 10, 19,
  and 24 are not pairwise relatively prime.

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Exercise 3.4

• Determine whether the integers in each of these
  sets are pairwise relatively prime.
• 11, 15, 19
• 14, 15, 21
• 12, 17, 31, 37
• 7, 8, 9, 11

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Find the GCD by Prime Factorization

• Another way to find the GCD of two integers is to
  use the prime factorizations of these integers.

If the prime factorizations of the integers a
and b are Then GCD (a, b) is given by
TIPS
min (a, b) represents the minimum of the two
integers a and b
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Example

• What is the greatest common divisor of 120 and
  500?
• Solution
• The prime factorization of 120 is
• The prime factorization of 500 is
• Hence,

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Exercise 3.4

• What are the greatest common divisors of these
  pairs of integers?
• ,
• ,
• ,
• ,
• ,

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3.5 LEAST COMMON MULTIPLE (LCM)
Lesson outcome Find the least common multiple
of two integers
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Least Common Multiple (LCM)

• DEFINITION
• If a, b, and k are in , and a k and b k
  , we say that k is a common multiple of a and b.
• The smallest k, call it c is called the least
  common multiple, or LCM, of a and b, and we write
  c LCM (a, b).

TIPS
The least common multiple of two numbers is the
smallest number (not zero) that is a multiple of
both.
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Example

• Find LCM(3,4).
• Solution
• Multiple of 3 3, 6, 9, 12, 15,
• Multiple of 4 4, 8, 12, 20, 24,
• LCM(3, 4) 12

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Exercise 3.5

• Find LCM(6,4).
• What is the least common multiple of 34 and 58?
• Find LCM(12,8).
• What is the least common multiple of 10 and 15?

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Find the LCM by Prime Factorization

• Another way to find the LCM of two integers is to
  use the prime factorizations of these integers.

If the prime factorizations of the integers a
and b are Then LCM (a, b) is given by
TIPS
max(a, b) represents the maximum of the two
integers a and b
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Example

• What is the least common multiple of 120 and 500?
• Solution
• The prime factorization of 120 is
• The prime factorization of 500 is
• Hence,

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Exercise 3.5

• What are the least common multiple of these pairs
  of integers?
• ,
• ,
• ,
• ,
• ,

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Obtain the LCM from the GCD

• Theorem
• If a and b are two positive integers, then
• ab GCD(a, b) LCM(a, b)

EXAMPLE Given GCD(190,34) 2. Determine
LCM(190,34). Solution
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Exercise 3.5

• Given GCD(105,30) 15. Determine LCM(105,30).
• Given GCD(540,504) 36.
• Determine LCM(540,504).
• Given GCD(12,30) 6. Determine LCM(12,30).

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3.6 EUCLEDIAN ALGORITHM
Lesson outcome Use the Euclidean Algorithm to
find the Greatest Common Divisor of two integers
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Euclidean Algorithm

• - A systematic procedure for finding GCD(a, b)
  is Euclidean Algorithm. This procedure is based
  on the following result (Lemma) about GCD and the
  division algoithm below.

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Example

• Determine GCD(190,34) by using Euclidean
  algorithm.
• Solution
• a 190 and b 34. By using the Euclidean
  algorithm, we
• divide 190 by 34 190 5(34) 20
• divide 34 by 20 34 1(20) 14
• divide 20 by 14 20 1(14) 6
• divide 4 by 6 14 2(6) 2
• divide 6 by 2 6 3(2)
  0,
• so GCD(190,34) 2, the last of nonzero divisors.

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Exercise 3.6

• Use the Euclidean algorithm to find
• GCD(414,662)
• GCD(120,23)
• GCD(12, 18)
• GCD(111, 201)
• GCD(1001, 1331)
• GCD(12345, 54321)
• GCD(1000, 5040)
• GCD(9888, 6060)

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Extended Euclidean Algorithm

• The GCD of two integers can be expressed as the
  following theorem

Linear combination
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Example

• Express gcd(252, 198) 18 as a linear
  combination of 252 and 198.
• Solution
• To show that gcd(252, 198) 18, the Euclidean
  algorithm uses these divisions
• 252 1(198) 54
• 198 3(54) 36
• 54 1(36) 18
• 36 2(18)

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Example

• Express gcd(252, 198) 18 as a linear
  combination of 252 and 198.
• Solution (continue)
• 54 252 1(198)
• 36 198 3(54)
• 198 3252-1(198)
• 198 3(252) 3(198)
• 4(198) 3(252)
• 18 54 1(36)
• 252 1(198) -14(198)-3(252)
• 252 198 4(198) 3(252)
• 4(252) 5(198) s 4, t -5

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Exercise 3.6

• Express gcd(190, 34) 2 as a linear combination
  of 190 and 34.
• Express gcd(120, 23) 1 as a linear combination
  of 120 and 23.

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Exercise 3.6

• Express the greatest common divisor of each of
  these pairs of integers as a linear combination
  of these integers.
• 10, 11 f) 0, 223
• 21, 44 g) 123, 2347
• 36, 48 h) 3454, 4666
• 34, 55 i) 9999, 11111
• 117, 213

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3.7 MODULAR ARITHMETIC
Lesson outcome Apply the use of congruences and
modular arithmetic in hashing function,
pseudorandom numbers and cryptology.
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Motivation
In some situations we care only about the
remainder of an integer when it is divided by
some specified positive integer.
Example If the time is now 9 oclock, what time
will it be 100 hours from now? Solution Let n
24 and m 9 100 109. Then we have 109
4(24) 13. In 100 hours it will be 13 oclock.
Recall The Division Algorithm

• r m mod n 13 109 mod 24

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Example

• Evaluate these quantities

• 17 mod 3

b) 133 mod 9
solution
solution
c) 2004 mod 101
d) 29 mod 5
solution
solution
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Exercise 3.7

• Evaluate these quantities.
• 13 mod 3
• -97 mod 11
• 155 mod 19
• -221 mod 23
• 3 mod 6

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Congruent
m and r not congruent modulo n
m mod n r mod n
congruent to r modulo n
n is modulus
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Example

• Determine whether 17 is congruent to 5 modulo 6
• Determine whether 24 and 14 are congruent modulo
  6.

solution
solution
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Exercise 3.6

• Decide whether each of these integers is
  congruent to 5 modulo 17.
• 80
• 103
• -29
• -122

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Mod-n Function

• Example
• because 16 5(3) 1 and
• because 156 22(7) 2 and
• because 14 4(3) 2 and

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Exercise 3.6

• If f is the mod-7 function, compute each of the
  following.
• f (17)
• f (48)
• f (1207)
• f (130)
• f (93)
• f (169)

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Example

• If f is the mod-7 function, solve f (n) 2.

solution
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Exercise 3.6

• If g is the mod-6 function, solve each of the
  following.
• g (n) 3
• g (n) 1

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Example

• Solve the following congruences if possible.
  If no solution exists, explain why not.

solution
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Exercise 3.6

• Solve each of the following congruences if
  possible. If no solution exists, explain why not.

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Applications Of Congruences

• Hashing Functions
• The use of congruences to assign memory locations
  to computer files
• Pseudorandom Numbers
• The generation of pseudorandom numbers
• Cryptography
• Cryptosystems based on modular arithmetic

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1. Hashing Functions
MOTIVATION
The central computer of a university maintains
records for each student. How can memory
locations be assigned so that student records can
be retrieved quickly?
THE SOLUTION
Use the suitable Hashing Function

• Many different hashing functions are used so that
  files can be quickly located .
• One of the most common is the function
• where m is the number of available memory
  locations and k is a key which uniquely
  identifies each records.

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Example
Hashing Function
If the number of available memory locations are
111
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Exercise 3.6

• Which memory locations are assigned by the
  hashing function h (k) k mod 101 to the records
  of students with this ID numbers?
• 104578690
• 432222187
• 372201919
• 501338753

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Exercise 3.6

• A parking lot has 31 visitor spaces, numbered
  from 0 to 30.Visitors are assigned parking spaces
  using the hashing function h (k) k mod 31,
  where k is the number formed from the first three
  digits on their license plates.
• Which spaces are assigned by the hashing
  function to cars that have these first three
  digits on their license plates?
• 317, 918, 007, 100, 111, 310

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2. Pseudorandom Numbers
DEFINITION
Random numbers generated by systematic methods
that are not truly random
THE PROCEDURE
Use the Linear Congruential Method

• Choose four integers modulus m, multiplier a,
  increment c, and seed x0 with 2 a lt m, 0 c lt
  m , and 0 x0 lt m
• Generate a sequence of pseudorandom number xn
  with 0 xn lt m for all n, by successively
  using the congruence

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Example

• The sequence of pseudorandom numbers generated by
  choosing m 9, a 7, c 4, and x0 3 can be
  found as follows
• Because x9 x0 3, thus these sequence contains
  9 different numbers before repeating as below
• 3, 7, 8, 6, 1, 2, 0, 4, 5, 3, 7, 8, 6, 1, 2, 0,
  4, 5, 3,

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Exercise 3.6

• What sequence of pseudorandom numbers is
  generated using the linear congruential generator
  
• with seed 2?
  

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3. Cryptography
FOUNDER
Use by Julius Caesar to write a secret messages
by shifting each letter 3 letters forward in
alphabet.
ENCRYPTION
The process of making secret message
STEPS (Caesar cipher )

• Replace each letter by an integer from 0 to 25,
  based on its position in the alphabet.
• Replace each of these numbers p by f (p) (p
  3) mod 26
• Substitute the letters that correspond to these
  numbers.

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STEP 1 (Caesar Cipher )
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Example
74
Example
75
Exercise 3.6

• Encrypt the message DO NOT PASS GO by
  translating the letters into numbers, applying
  the encryption function given, and then
  translating the numbers back into letters.
• (the
  Caesar cipher)

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Exercise 3.6

• Decrypt these messages encrypted using the Caesar
  cipher.
• EOXH MHDQV
• WHVW WRGDB
• HDW GLP VXP
• L DP ILQH

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3.8 THE CHINESE REMINDER THEOREM
Lesson outcome Find the smallest nonnegative
integer that satisfies the given system of
congruences.
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Motivation
Puzzle
Asked by a Chinese mathematician Sun-Tsu in the
first century.
There are certain things whose number is unknown.
When divided by 3, the remainder is 2 when
divided by 5, the remainder is 3 and when
divided by 7, the remainder is 2. What will be
the number of things?

• This puzzle can be translated into the following
  question
• What are the solutions of the systems of
  congruences

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The Chinese Remainder Theorem
It follows that
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Example

• Solve the system

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Example Sun-Tsu Puzzle

• What are the solutions of the systems of
  congruences

82
Exercise 3.7

• In each case, find the smallest nonnegative
  integer that satisfies the given system of
  congruences.
• c)

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Thank You