# 5-1 NUMBER THEORY: DIVISIBILITY; PRIME & COMPOSITE NUMBERS 210 f8

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 5-1 NUMBER THEORY: DIVISIBILITY; PRIME & COMPOSITE NUMBERS 210 f8 Note: Integers are the w hole numbers and their negatives (additive inverses). While our text discusses only whole numbers, all these ideas extend to the negative integers as well as positive, so the statements can be inclusive. For all statements below, variables a, b, c, d, and k are whole numbers (or integers).* Def' n: For k,b integers k*b (k divides b) if, & only if, b = kc for some integer* c. If k b w e say " k is a divisor or factor of b" " b is a multiple of k" equivalent statements b is divisible by k If k is not a factor of b, w e say k does not divide b, and w rite k b. for example 4 20 because 20 = 5 4 eg 4 17 EG 3 *12 (or 12 is a multiple of 3) 6 is a factor of 24 Is 42 divisible by 3? because 12 = = = 7 6 & 6 = 3 2 so 42 = & 4 is a w hole number Fact: If a W then a*0 because 0 = a 0 (or Z) and 1*a because a = 1 a and a*a because a = a 1 Fact: If d*a, then d*ab EG 150 must be divisible by 3, because 15 is... Pf : if d*a then a = dc Think: If 3 15 then 15 = 3 5 ab = (dc)b = d(cb), Then 15 w must be 3 5 w so ab is a multiple of d PQED So 15w must be a multiple of 3. Fact: If a*b and b*c then a*c (" divides" is transitive) Pf : a*b and b*c c = bm and b = an, for some m,n Z [ by def'n of " *" ] bm = (an)m & (an)m = a(nm) [ assoc prop x ] so c = a(nm) and c = a(some integer) [ closure prop of mult.]...thus proving that a*c PQED COMMENT: This is almost the same fact as above. E.G. Since 3 15, and , therefore Fact: If d*a and d*b, then d*(a+ b) and d*(a b). Pf: If d*a and d*b, then a = dc and b = de, so a+ b = dc + de = d(c+ e) PQED The second follow s immediately from the first, since a b = a + ( b). Put another way: if a and b are both multiples of d, then a+ b is also a multiple of d. For example: 7000 and 21 are both multiples of 7, therefore 7021 must be a multiple of 7. Fact: If d*a and d b, then d (a+ b). Pf: If d*a and d*(a+ b) then d must divide b since d*a d* a so d*((a+ b) + ( a)) and (a+ b) + ( a) = b. PQED Put another way: If a is a multiple of d, and b is not, then a+ b cannot be a multiple of d. For example: 90 is a multiple of 3, 2 is not. Therefore, 92 cannot be a multiple of 3. (We all know the multiples of 3 are spaced 3 units apart!!! ) Notice the tw o above facts together say: If d*a then d*(a+ b) iff d*b....that is: If d is already know n to divide a, then to see if d*(a+ b), just test w hether d*b.

2 TESTS for DIVISIBILITY One way to determine whether a divides b is to attempt to divide b by a. How ever, there are a number of " tests" for divisibility by certain values. We are expected to know these. DIV ISIBILITY TEST FOR 2 : Is the units digit even? (i.e. is the units digit divisible by 2)? This fact is often observed by children who learn to count by twos. Eg. It is obvious to all of us that 3574 is divisible by 2 because 2*4. This shortcut w orks because 3574 = is a multiple of 2, and 2*10 (so 2*357x10... ie 2*3570). In general, if u is the units digit, and v represents the rest of the value of a number, then the number is actually v+ u and v is a multiple of 10. Since 2*10 and 10*v, w e know automatically, w ithout inspecting v, that 2*v. If 2 also divides u, the units digit, then 2*(v+ u)....because of On the other hand, if 2 u, since w e know 2*v, 2 (v+ u)....because of DIVISIBILITY TEST FOR 5: Is the units digit 0 or 5? (i.e. is the units digit divisible by five)? This fact is genereally realized by children who learn to count by fives. Eg. That is divisible by 5 is obvious to anyone who has counted by fives. This shortcut works for the same reason that the test for divisibility by 2 works... because = and = 3117x10 and 5*10, so 5* In general, if u represents the units digit, and v represents the rest of the value of a number, then v is a multiple of 10 and the number is v+ u. Since 5*10 and 10*v, w e know automatically, w ithout inspecting v, that 5*v. Thus 5*(v+ u) if and only if 5*u....because of multiple of 10 DIVISIBILITY TEST FOR 10: Is the units digit 0? (divisible by ten)? E.g is obviously NOT divisible by 10. This w orks because = and = 3457 x 10 so 10*34570 but In general, if the units digit of a number is u, and v represents the remaining value of a number, then the entire value of the number is v+ u (just as = ). v is a multiple of 10. So w e need check only u (because of ). Since 10*v, 10*(v+ u) iff 10*u. DIVISIBILITY TEST FOR 4: Is the number represented by the two right-most digits divisible by four? E.g is divisible by 4 since 4* is not divisible by 4 since Why it w orks: = = , and 4*100, so 4*56700 automatically. Since 4 also divides 12, 4 must divide Similarly, = Since 4*56700 and 4 18, it follows that In general, if " tu" represents the number represented by the tens & units digits, and v represents the remaining value of a number, then the number is v + "tu" and v is a multiple of 100. Since 100 is divisible by 4, 4*v. If 4 also divides " tu" then 4 must divide v + "tu". (because of ) If 4 "tu" then 4 cannot divide v + "tu". (because of ) DIVISIBILITY TEST FOR 8: check the last digits... 3 E.g is divisible by 8 since 8*. 120 This w orks because...

3 M ORE TESTS for DIVISIBILITY DIVISIBILITY TEST FOR 3: Is the SUM OF THE DIGITS divisible by three? Eg is divisible by 3 since = 21 and 21 is divisible by 3. This test w orks because = = 8 ( ) + 6(99 + 1) + 5 (9+ 1) = = ( ) Multiple of 9, thus of 3 21, a multiple of 3 In general: The value of each digit in a numeral (in our numeration system) is its face value times a pow er of ten. But each pow er of ten is exactly 1 + (a multiple of nine). 3*9 so 3*(any multiple of 9). Thus 3 divides the given number if, and only if, 3 divides the sum of the digits, as seen in the example above. The same argument demonstrates the reason behind the divisibility test for nine, w hich follow s... DIVISIBILITY TEST FOR 9: Is the SUM OF THE DIGITS divisible by nine? EG: because the sum of the digits is 21, not a multiple of because = because = 27, a multiple of 9. This w orks because... (see above). DIVISIBILITY TEST FOR 11: Is the "ALTERNATING SUM" OF THE DIGITS divisible by eleven? Eg is divisible by 11 since = 0 and 0 is divisible by 11***. This works because = = 5 (11 1) (11 1) (11 1) (11 1) + 8 (11 1) (11 1) = 5 (11j 1) + 6 (11k+ 1) + 2 (11m 1) + 2 (11n+ 1) + 8 (11 1) = 5 (11j) + 6 (11k) + 2 (11m) + 2 (11n) + 8 (11) + ( ) As we can see, 11 divides the first terms, so divisibility of by 11 depends on the divisibility of the last part the alternating sum of the digits, (Q: Do w e start w ith + or? A: Try both & see: ) For the follow ing, think of factor trees. Eg 36 What numbers divide (are factors of) 36? 4 9 1, 2, 3, 4, 6, 9, 12, 18, DIVISIBILITY "TEST" FOR 6: DIVISIBILITY TEST FOR 12: True or false: If a number is divisible by 2 and by 9, then it is divisible by 18. (A) True or false: If a number is divisible by 3 and by 6, then it is divisible by 1 8. (B) (Statement A above is true. To see that Statement B is false, consider the number 6. Or 1 2. Or 2 4. Or 3 0. ) * * * As noted previously, if a is any w hole number (or any integer), then a 0. ( because a 0 = 0 ) That is, 0 is a multiple of EVERY whole number. Put another w ay: 0 belongs on the list of multiples of 2. (, 6, 4, 2, 0, 2, 4, 6, ) 0 belongs on the list of multiples of 3. (, 9, 6, 3, 0, 3, 6, 9, ) 0 belongs on the list of multiples of any w hole number a. (, 3a, 2a, a, 0, m, 2m, 3m, )

4 5-3 PRIME NUMBERS 210 f8 Def'n: A PRIM E NUM BER is a natural number that has EXACTLY TWO natural number divisors 1 & itself. A COMPOSITE number is a natural number that has natural number divisors other than 1 and itself. Eg. 5 is a prime number, divisible by 1 and by 5, and no other elements of N. 6 is composite, since it is divisible by 2 and 3, in addition to 1 and 6. 1 is neither prime nor composite! ( 1 is called a UNIT. ) Prime or composite? 123 is (You can probably tell right aw ay, because 123 is clearly divisible by... ) 223 is In testing whether 223 is prime, must we test for divisibility...by 2? 3? 4? 5? 6? 7? 8?...by 19? because 3 is not even because = 7 Do w e need to test for divisibility by 4? No, if 223 were divisible by 4, it would be divisible by because... Do we need to test for divisibility by 6? No, if 223 were divisible by 6, it would be... So we need to test for divisibility by only primes only, right? because 223 = because 223 = because 223 = hmmm... do w e have to test every prime up to 223? Well, it turns out NOT. We have to test only up to the square root of 223, which is less than 17, in fact less than 15. WHY?... Consider the non-trivial* factor pairs of = 2 36 = 3 24 * Every w hole number w is the product of 1 & itself w = 1 w. This is called the trivial factorization...and w and 1 are called trivial factors. = 4 18 = 6 12 = 8 9 In this example n = 72. In the following, think n= 72: If a is a factor of n, then n/a = c is a factor of n. That is: n = a c. If a is greater than n, then c is smaller than n. (Eg 18 > 7 2, but then c = 3, much less than 7 2. Not both of a and c can be greater than n (if both > n, then a c > n n a contradiction of the fact that n = a c). So if n has any factors other than n and 1, there must be at least one n. If none exist, then n has no factors other than the trivial factors, 1 and n and must be a prime. To test whether a given number "n" is prime, test for divisibility by up to THERE ARE INFINITELY M ANY PRIM ES. [see text! It does a great job of showing you that if you think you have them all, then there must be one more that is greater than any in your list... therefore, you can never have them all! ] A composite number may be expressed as a product of two or more non-trivial factors For any n, there exists a run of n consecutive composite numbers: Let m = (n+ 1). Then m+ 2 is composite because 2*m and 2*2; m+ 3 is composite because... Similarly, m+ k is composite for k = 2,3,4,...,n+ 1. PQED

5 5-3 More on PRIME NUMBERS 210 f8 Reminder: A composite number may be expressed as a product of tw o or more non-trivial factors (the trivial factors are 1 and the number itself). Such an expression is called a FACTORIZATION of the number. Eg. The composite number 12 may be expressed as or as or as = 3 12 = 4 9 =... = 6 6 = 2 18 = = = 2 30 =... = Intuitively w e can see that a composite number can be factored & if the factors are not all prime, they can be factored again... until the factors are all primes, no further factorization is possible: we call this the PRIME FACTORIZATION of the number. There is only one prime factorization of the number, as stated in the: FUNDAMENTAL THEOREM OF ARITHM ETIC: Every natural number > 1 has a unique prime factorization. Example: Find the prime factorization of 132. Tree: = = = Find the prime factorizations of Up for a challenge? Find n and m such that = n 3 m Think prime! Find the prime factorizations of these. (You might want to use a calculator to save some time.) 70,224 70,

6 5-5 GCF & LCM: Greatest Common Factor and Least Common Multiple 210 f8 If a natural number d divides every one of a set of integers, then d is a common factor or common divisor of those integers. Eg. 72 and 60 have common factors: 1, 2, 3, 4, 6, and 12. Eg. 22 and 27 have common factors: 1 Eg. 12, 30, and 75 have common factors: 1 and 3 The largest or maximum common factor of tw o w hole numbers is called the GREATEST COMMON FACTOR, or GCF, of the numbers, called GCF(a,b)...also called the greatest common divisor, or GCD. Eg. GCF( 72, 60) = 12 GCF( 22, 27) = 1 GCF( 12, 30, 75) = 3 If the GCF of tw o numbers is 1, the numbers are said to be relatively prime. Eg. 22 and 27 are relatively prime; they have no factors, other than 1, in common. Three Methods for obtaining GCF(a,b) M1. List the factors of the numbers: Eg. GCF(72,60) factors of 72: {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72} factors of 60: {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60} Find the common factors: {1, 2, 3, 4, 6, 12} Find the greatest: M2. Write the prime factorization of the numbers: 72 = = Build the largest possible factor of both by taking the low est pow er of each common prime factor: GCF( 72, 60) = GCF( 2 3, 2 3 5) = 2 3 = 12 Fill in the Exponents! GCF( 336, 240, 990) = GCF( 2 3 7, 2 3 5, ) = = GCF( 5040, 960) = GCF(, ) = = M3. Euclidean Algorithm Fact: For any a,b in N with a > mb for some m in N, GCF(a,b) = GCF(b,a b) = GCF(b,a mb). Pf: If d = GCF(a,b) then d*a and d*b, so d*(a b) and d is in the set of common factors of a b and b. Conversely, if d*(a b) and d*b then d*((a b)+ b), that is d*a, so d is in the set of common factors of a and b. Thus the set of common factors of (a,b) and the set of common factors of (a b,b) are identical. Induction on m N extends this to GCF(a mb,b). Notice that a mb is the remainder obtained by subtracting m " b's" from a. So if mb is the largest multiple of b that is a, then a mb must be the remainder obtained w hen a is divided by b. This implies that GCF(a,b) = GCF(r,b) w here r is the remainder obtained when a is divided by b! Eg. GCF(30,12) = GCF(6,12) = 6. Eg. GCF(72,30) = GCF(12,30)... but GCF(30,12) = GCF(6,12) = 6... So GCF(72,30) = 6. We have just discovered the secret of the Euclidean algorithm for finding the GCF of two numbers: Divide and use the remainder... Eg. GCF(676,182): 676 = GCF(676,182) = GCF(182,130) What 182 = GCF(182,130) = GCF(130,52)...Why you do 130 = GCF(130,52) = GCF(52,26) it w orks. 52 = GCF(52,26) = 26 Note: To use the Euclidean algorithm for more than two numbers: find the GCF of any two; then find the GCF of that result w ith the next value, or w ith the GCF of another pair; etc. Each of the above methods (M1, M2, M3) has advantages. What is an advantage of each method?

7 5-5 GCF & LCM: Least Common Multiple 210 f8 If a natural number d is a multiple of every one of a particular set of integers, then d is a common multiple of those integers. The smallest or minimum common multiple of two whole numbers is called the LEAST COM M ON M ULTIPLE, or LCM, of the numbers, notated LCM(a,b,...). Eg. multiples of 12 = {12, 24, 36, 48, 60, 72, 84, 96, 108,... } multiples of 16 = {16, 32, 48, 64, 80, 96, 112, 128, 144,... } 12 and 16 have common multiples: {48, 96, 144,...} LCM( 12, 16) = 48 Eg. LCM( 12, 60 ) = 60 LCM( 24, 15 ) = 120 LCM( 6, 21, 10 ) = 210 Three Methods for obtaining LCM(a,b) M1. List the multiples of the numbers: Eg. LCM( 72, 60 ) multiples of 72: { 72, 144, 216, 288, 360, 432, 504, 576,... } multiples of 60: { 60, 120, 180, 240, 300, 360, 420, 480, 540,...} Find the common multiples: { 360, 720, 10800,... } Find the least (common mulitple): 360 M2. Write the prime factorization of the numbers: = = Build the least possible multiple of both by taking the highest pow er of each prime factor: LCM( 72, 60 ) = LCM( 2 3, ) = = 360 LCM(12,14,30) = LCM( 2 2 3, 2 7, ) = = LCM(18,60,54) = LCM( 2 3, 2 3 5, 2 3 ) = = LCM(336,240,990) = LCM( 2 3 7, 2 3 5, ) = = Fill in the Exponents! M3. Use the GCF Fact: For any a,b in N the product of their LCM & GCF is ab: GCF(a,b) LCM(a,b) = ab. r q Pf: Relies on the fact that if a prime appears in the factorization of a or b, then a = p m and b = p n r q and ab = p p m n. GCF contains p to the pow er min(r,q) and LCM contains p to the pow er max(r,q). min(r,q) + max (r,q) = r + q, the pow er on p in the factorization of ab. In short, all the pow ers of primes in the factorizations of a and b end up in either the GCF or the LCM... Multiplying the tw o puts them altogether, reconstituting ab. Thus we may use the Euclidean algorithm LCM(a,b) = a b to find the LCM of two numbers, since GCF(a,b) Eg. LCM( 72, 60 ) = / GCF(72,60) = 72 60/12 = 6 60 = 360 Eg. LCM( 676, 182 ) = Note: To find the LCM of more than two numbers via method (3), find the LCM of pairs, then find the LCM of the results (until every original number has been included). Something to think about! Given the prime factorization of a number, how can w e find the number of factors of the number? Here are some questions that may lead to an answ er: 3 8 = 2 W hat numbers divide 8? How many factors has 8? (Ans: four: 1, 2, 4, 8) What options do you have w hen "building" a factor of 8? (Ans: these four: 2, 2, 2, 2 ) = = 2 3 What numbers divide 36? (2? 3? 5? 8?) What goes into a factor of 36; how can you " build" a factor of 36? What choices do you have to make w hen building a factor of 36? How many factors of 36 exist in N? (Ans: 9) Write the prime factorization of 72. Write the prime factorizations of all the factors of 72 (these w ere listed on page 5-5). What makes up the factors of 72? How many different w ays can you build a number w hich divides 72? (If you are " building" a factor of 72, can you use 5? [no]...can you use 2? 4? 8? 16? no 2' s at all?...

### Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University

Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called

### Homework 5 Solutions

Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which

### Algebra for Digital Communication

EPFL - Section de Mathématiques Algebra for Digital Communication Fall semester 2008 Solutions for exercise sheet 1 Exercise 1. i) We will do a proof by contradiction. Suppose 2 a 2 but 2 a. We will obtain

### 17 Greatest Common Factors and Least Common Multiples

17 Greatest Common Factors and Least Common Multiples Consider the following concrete problem: An architect is designing an elegant display room for art museum. One wall is to be covered with large square

### Prime Factorization, Greatest Common Factor (GCF), and Least Common Multiple (LCM)

Prime Factorization, Greatest Common Factor (GCF), and Least Common Multiple (LCM) Definition of a Prime Number A prime number is a whole number greater than 1 AND can only be divided evenly by 1 and itself.

### 18. [Multiples / Factors / Primes]

18. [Multiples / Factors / Primes] Skill 18.1 Finding the multiples of a number. Count by the number i.e. add the number to itself continuously. OR Multiply the number by 1, then 2,,, 5, etc. to get the

### Lecture 1: Elementary Number Theory

Lecture 1: Elementary Number Theory The integers are the simplest and most fundamental objects in discrete mathematics. All calculations by computers are based on the arithmetical operations with integers

### Prime Numbers. Chapter Primes and Composites

Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are

### FACTORING OUT COMMON FACTORS

278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the

### MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### Math 201: Homework 7 Solutions

Math 201: Homework 7 Solutions 1. ( 5.2 #4) (a) The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. (b) The factors of 81 are 1, 3, 9, 27, and 81. (c) The factors of 62 are 1, 2, 31, and

### Session 6 Number Theory

Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple

### CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

### Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,

### CISC - Curriculum & Instruction Steering Committee. California County Superintendents Educational Services Association

CISC - Curriculum & Instruction Steering Committee California County Superintendents Educational Services Association Primary Content Module IV The Winning EQUATION NUMBER SENSE: Factors of Whole Numbers

### Add: 203.602 + 3.47 203.602 = 200 + 3 + 6/10 + 0/100 + 2/1000 + 3.47 + 3 + 4/10 + 7/100

9-1 DECIM ALS Their PLACE in the W ORLD Math 10 F8 Definition: DECIM AL NUM BERS If x is any w hole number, x. d1 d d3 d 4... represents the number: x + d 1/10 + d /100 + d 3/1000 + d 4/10000 + d 5/100000...

### Primes. Name Period Number Theory

Primes Name Period A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following exercise: 1. Cross out 1 by Shading in

### 8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

### 15 Prime and Composite Numbers

15 Prime and Composite Numbers Divides, Divisors, Factors, Multiples In section 13, we considered the division algorithm: If a and b are whole numbers with b 0 then there exist unique numbers q and r such

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

### 6.1 The Greatest Common Factor; Factoring by Grouping

386 CHAPTER 6 Factoring and Applications 6.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### 2 The Euclidean algorithm

2 The Euclidean algorithm Do you understand the number 5? 6? 7? At some point our level of comfort with individual numbers goes down as the numbers get large For some it may be at 43, for others, 4 In

### Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.

Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole

### Number Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may

Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition

### Homework until Test #2

MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

### b) Find smallest a > 0 such that 2 a 1 (mod 341). Solution: a) Use succesive squarings. We have 85 =

Problem 1. Prove that a b (mod c) if and only if a and b give the same remainders upon division by c. Solution: Let r a, r b be the remainders of a, b upon division by c respectively. Thus a r a (mod c)

### Chapter 6. Number Theory. 6.1 The Division Algorithm

Chapter 6 Number Theory The material in this chapter offers a small glimpse of why a lot of facts that you ve probably nown and used for a long time are true. It also offers some exposure to generalization,

### Number Theory Hungarian Style. Cameron Byerley s interpretation of Csaba Szabó s lectures

Number Theory Hungarian Style Cameron Byerley s interpretation of Csaba Szabó s lectures August 20, 2005 2 0.1 introduction Number theory is a beautiful subject and even cooler when you learn about it

### SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

### There are 8000 registered voters in Brownsville, and 3 8. of these voters live in

Politics and the political process affect everyone in some way. In local, state or national elections, registered voters make decisions about who will represent them and make choices about various ballot

### The Euclidean Algorithm

The Euclidean Algorithm A METHOD FOR FINDING THE GREATEST COMMON DIVISOR FOR TWO LARGE NUMBERS To be successful using this method you have got to know how to divide. If this is something that you have

### Further linear algebra. Chapter I. Integers.

Further linear algebra. Chapter I. Integers. Andrei Yafaev Number theory is the theory of Z = {0, ±1, ±2,...}. 1 Euclid s algorithm, Bézout s identity and the greatest common divisor. We say that a Z divides

### Kevin James. MTHSC 412 Section 2.4 Prime Factors and Greatest Comm

MTHSC 412 Section 2.4 Prime Factors and Greatest Common Divisor Greatest Common Divisor Definition Suppose that a, b Z. Then we say that d Z is a greatest common divisor (gcd) of a and b if the following

### 9. The Pails of Water Problem

9. The Pails of Water Problem You have a 5 and a 7 quart pail. How can you measure exactly 1 quart of water, by pouring water back and forth between the two pails? You are allowed to fill and empty each

### Unique Factorization

Unique Factorization Waffle Mathcamp 2010 Throughout these notes, all rings will be assumed to be commutative. 1 Factorization in domains: definitions and examples In this class, we will study the phenomenon

### Exponents, Factors, and Fractions. Chapter 3

Exponents, Factors, and Fractions Chapter 3 Exponents and Order of Operations Lesson 3-1 Terms An exponent tells you how many times a number is used as a factor A base is the number that is multiplied

### Chapter 3. if 2 a i then location: = i. Page 40

Chapter 3 1. Describe an algorithm that takes a list of n integers a 1,a 2,,a n and finds the number of integers each greater than five in the list. Ans: procedure greaterthanfive(a 1,,a n : integers)

### Integers and division

CS 441 Discrete Mathematics for CS Lecture 12 Integers and division Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Symmetric matrix Definition: A square matrix A is called symmetric if A = A T.

### k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

### It is time to prove some theorems. There are various strategies for doing

CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it

### In the above, the number 19 is an example of a number because its only positive factors are one and itself.

Math 100 Greatest Common Factor and Factoring by Grouping (Review) Factoring Definition: A factor is a number, variable, monomial, or polynomial which is multiplied by another number, variable, monomial,

### Practice Problems for First Test

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.-

### Course notes on Number Theory

Course notes on Number Theory In Number Theory, we make the decision to work entirely with whole numbers. There are many reasons for this besides just mathematical interest, not the least of which is that

### 5.1 FACTORING OUT COMMON FACTORS

C H A P T E R 5 Factoring he sport of skydiving was born in the 1930s soon after the military began using parachutes as a means of deploying troops. T Today, skydiving is a popular sport around the world.

### The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.

The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,

### (0, 0) : order 1; (0, 1) : order 4; (0, 2) : order 2; (0, 3) : order 4; (1, 0) : order 2; (1, 1) : order 4; (1, 2) : order 2; (1, 3) : order 4.

11.01 List the elements of Z 2 Z 4. Find the order of each of the elements is this group cyclic? Solution: The elements of Z 2 Z 4 are: (0, 0) : order 1; (0, 1) : order 4; (0, 2) : order 2; (0, 3) : order

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### MACM 101 Discrete Mathematics I

MACM 101 Discrete Mathematics I Exercises on Combinatorics, Probability, Languages and Integers. Due: Tuesday, November 2th (at the beginning of the class) Reminder: the work you submit must be your own.

### Grade 6 Math Circles March 10/11, 2015 Prime Time Solutions

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Lights, Camera, Primes! Grade 6 Math Circles March 10/11, 2015 Prime Time Solutions Today, we re going

### ( ) FACTORING. x In this polynomial the only variable in common to all is x.

FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated

### 4. Number Theory (Part 2)

4. Number Theory (Part 2) Terence Sim Mathematics is the queen of the sciences and number theory is the queen of mathematics. Reading Sections 4.8, 5.2 5.4 of Epp. Carl Friedrich Gauss, 1777 1855 4.3.

### In a triangle with a right angle, there are 2 legs and the hypotenuse of a triangle.

PROBLEM STATEMENT In a triangle with a right angle, there are legs and the hypotenuse of a triangle. The hypotenuse of a triangle is the side of a right triangle that is opposite the 90 angle. The legs

### An Introduction to Number Theory Prime Numbers and Their Applications.

East Tennessee State University Digital Commons @ East Tennessee State University Electronic Theses and Dissertations 8-2006 An Introduction to Number Theory Prime Numbers and Their Applications. Crystal

### COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13

COMPASS Numerical Skills/Pre-Algebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre

### MATH 22. THE FUNDAMENTAL THEOREM of ARITHMETIC. Lecture R: 10/30/2003

MATH 22 Lecture R: 10/30/2003 THE FUNDAMENTAL THEOREM of ARITHMETIC You must remember this, A kiss is still a kiss, A sigh is just a sigh; The fundamental things apply, As time goes by. Herman Hupfeld

Answers Investigation Applications. a. Divide 24 by 2 to see if you get a whole number. Since 2 * 2 = 24 or 24, 2 = 2, 2 is a factor. b. Divide 29 by 7 to see if the answer is a whole number. Since 29,

### Squares and Square Roots

SQUARES AND SQUARE ROOTS 89 Squares and Square Roots CHAPTER 6 6.1 Introduction You know that the area of a square = side side (where side means the length of a side ). Study the following table. Side

### Grade 7/8 Math Circles Fall 2012 Factors and Primes

1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Factors and Primes Factors Definition: A factor of a number is a whole

### s = 1 + 2 +... + 49 + 50 s = 50 + 49 +... + 2 + 1 2s = 51 + 51 +... + 51 + 51 50 51. 2

1. Use Euler s trick to find the sum 1 + 2 + 3 + 4 + + 49 + 50. s = 1 + 2 +... + 49 + 50 s = 50 + 49 +... + 2 + 1 2s = 51 + 51 +... + 51 + 51 Thus, 2s = 50 51. Therefore, s = 50 51. 2 2. Consider the sequence

### Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.

Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,

### Today s Topics. Primes & Greatest Common Divisors

Today s Topics Primes & Greatest Common Divisors Prime representations Important theorems about primality Greatest Common Divisors Least Common Multiples Euclid s algorithm Once and for all, what are prime

### 5-4 Prime and Composite Numbers

5-4 Prime and Composite Numbers Prime and Composite Numbers Prime Factorization Number of Divisorss Determining if a Number is Prime More About Primes Prime and Composite Numbers Students should recognizee

### 6.2 Permutations continued

6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of

### Using the ac Method to Factor

4.6 Using the ac Method to Factor 4.6 OBJECTIVES 1. Use the ac test to determine factorability 2. Use the results of the ac test 3. Completely factor a trinomial In Sections 4.2 and 4.3 we used the trial-and-error

1. sum the answer when you add Ex: 3 + 9 = 12 12 is the sum 2. difference the answer when you subtract Ex: 17-9 = 8 difference 8 is the 3. the answer when you multiply Ex: 7 x 8 = 56 56 is the 4. quotient

### DigitalCommons@University of Nebraska - Lincoln

University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-1-007 Pythagorean Triples Diane Swartzlander University

### Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and

Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study

### CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

### Prime Numbers A prime number is a whole number, greater than 1, that has only 1 an itself as factors.

Prime Numbers A prime number is a whole number, greater than 1, that has only 1 an itself as factors. Composite Numbers A composite number is a whole number, greater than 1, that are not prime. Prime Factorization

Chapter 2 Remodulization of Congruences Proceedings NCUR VI. è1992è, Vol. II, pp. 1036í1041. Jeærey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department

### Mathematical induction & Recursion

CS 441 Discrete Mathematics for CS Lecture 15 Mathematical induction & Recursion Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Proofs Basic proof methods: Direct, Indirect, Contradiction, By Cases,

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

### Factoring Trinomials: The ac Method

6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For

### Mathematics of Cryptography Part I

CHAPTER 2 Mathematics of Cryptography Part I (Solution to Odd-Numbered Problems) Review Questions 1. The set of integers is Z. It contains all integral numbers from negative infinity to positive infinity.

### = 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

### Pre-Algebra Homework 2 Factors: Solutions

Pre-Algebra Homework 2 Factors: Solutions 1. Find all of the primes between 70 and 100. : First, reject the obvious non-prime numbers. None of the even numbers can be a prime because they can be divided

### FACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1

5.7 Factoring ax 2 bx c (5-49) 305 5.7 FACTORING ax 2 bx c In this section In Section 5.5 you learned to factor certain special polynomials. In this section you will learn to factor general quadratic polynomials.

### SPECIAL PRODUCTS AND FACTORS

CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

### Unit 1 Review Part 1 3 combined Handout KEY.notebook. September 26, 2013

Math 10c Unit 1 Factors, Powers and Radicals Key Concepts 1.1 Determine the prime factors of a whole number. 650 3910 1.2 Explain why the numbers 0 and 1 have no prime factors. 0 and 1 have no prime factors

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

### SOLUTIONS FOR PROBLEM SET 2

SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### SUM OF TWO SQUARES JAHNAVI BHASKAR

SUM OF TWO SQUARES JAHNAVI BHASKAR Abstract. I will investigate which numbers can be written as the sum of two squares and in how many ways, providing enough basic number theory so even the unacquainted

### MTH 382 Number Theory Spring 2003 Practice Problems for the Final

MTH 382 Number Theory Spring 2003 Practice Problems for the Final (1) Find the quotient and remainder in the Division Algorithm, (a) with divisor 16 and dividend 95, (b) with divisor 16 and dividend -95,

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### Solutions to Practice Problems

Solutions to Practice Problems March 205. Given n = pq and φ(n = (p (q, we find p and q as the roots of the quadratic equation (x p(x q = x 2 (n φ(n + x + n = 0. The roots are p, q = 2[ n φ(n+ ± (n φ(n+2

### 1.3 Polynomials and Factoring

1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.

### 4. FIRST STEPS IN THE THEORY 4.1. A

4. FIRST STEPS IN THE THEORY 4.1. A Catalogue of All Groups: The Impossible Dream The fundamental problem of group theory is to systematically explore the landscape and to chart what lies out there. We

### Factoring, Solving. Equations, and Problem Solving REVISED PAGES

05-W4801-AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring

### NUMBER THEORY AMIN WITNO

NUMBER THEORY AMIN WITNO ii Number Theory Amin Witno Department of Basic Sciences Philadelphia University JORDAN 19392 Originally written for Math 313 students at Philadelphia University in Jordan, this

### CSCI 246 Class 5 RATIONAL NUMBERS, QUOTIENT REMAINDER THEOREM

CSCI 246 Class 5 RATIONAL NUMBERS, QUOTIENT REMAINDER THEOREM Quiz Questions Lecture 8: Give the divisors of n when: n = 10 n = 0 Lecture 9: Say: 10 = 3*3 +1 What s the quotient q and the remainder r?

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

### Fractions and Decimals

Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first

### Chapter 3: Elementary Number Theory and Methods of Proof. January 31, 2010

Chapter 3: Elementary Number Theory and Methods of Proof January 31, 2010 3.4 - Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem Quotient-Remainder Theorem Given