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4 Chapter Number Theory Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

4-1 Divisibility Divisibility Divisibility Rules Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Divisibility A whole number is even if it has a remainder of 0 when divided by 2; it is odd otherwise. We say that “3 divides 18”, written 3 | 18, because the remainder is 0 when 18 is divided by 3. Likewise, “b divides a” can be written b | a. We say that “3 does not divide 25”, written , because the remainder is not 0 when 25 is divided by 3. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Divisibility In general, if a is a whole number and b is a non-zero whole number, we say that a is divisible by b, or b divides a if and only if the remainder is 0 when a is divided by b. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Definition If a and b are any whole number, then b divides a, written b | a, if, and only if, there is a unique whole number q such that a = bq. If b | a, then b is a factor or a divisor of a, and a is a multiple of b. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4-1 Classify each of the following as true or false. a. −3 | 12 True b. 0 | 2 False c. 0 is even. True d. True e. For all whole numbers a, 1 | a. True f. For all non-zero whole numbers a, a2 | a5. True g. 3 | 6n for all whole numbers n. True Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example (continued) h. True g. 0 | 0 False Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Properties of Division
For any whole numbers a and d, if d | a, and n is any whole number, then d | na. In other words, if d is a factor of a, then d is a factor of any multiple of a. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Properties of Division
For any whole numbers a, b, and d, d ≠ 0, a. If d | a, and d | b, then d | (a + b). b. If d | a, and , then c. If d | a, and d | b, then d | (a − b). d. If d | a, and , then Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4-2 Classify each of the following as true or false, where x, y, and z are whole numbers. a. If 3 | x and 3 | y, then 3 | xy. True b. If 3 | (x + y), then 3 | x and 3 | y. False c. False Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Divisibility Rules Sometimes it is useful to know if one number is divisible by another just by looking at it. For example, to check the divisibility of 1734 by 17, we note that 1734 = We know that 17 | 1700 because 17 | 17 and 17 divides any multiple of 17. Furthermore, 17 | 34; therefore, we conclude that 17 | 1734. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Divisibility Rules Another method to check for divisibility is to use the integer division button on a calculator. INT ÷ Press the following sequence of buttons: INT ÷ 1 7 3 4 = to obtain the display Q R Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Divisibility Tests A whole number is divisible by 2 if and only if its units digit is divisible by 2. A whole number is divisible by 5 if and only if its units digit is divisible by 5, that is if and only if the units digit is 0 or 5. A whole number is divisible by 10 if and only if its units digit is divisible by 10, that is if and only if the units digit is 0. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Divisibility Tests A whole number is divisible by 4 if and only if the last two digits of the number represent a number divisible by 4. A whole number is divisible by 8 if and only if the last three digits of the whole number represent a number divisible by 8. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4-4a Determine whether 97,128 is divisible by 2, 4, and 8. 2 | 97,128 because 2 | 8. 4 | 97,128 because 4 | 28. 8 | 97,128 because 8 | 128. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4-4b Determine whether 83,026 is divisible by 2, 4, and 8. 2 | 83,026 because 2 | 6. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Divisibility Tests A whole number is divisible by 3 if and only if the sum of its digits is divisible by 3. A whole number is divisible by 9 if and only if the sum of the digits of the whole number is divisible by 9. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4-6a Determine whether 1002 is divisible by 3 and 9. Because = 3 and 3 | 3, 3 | 1002. Because Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4-6b Determine whether 14,238 is divisible by 3 and 9. Because = 18 and 3 | 18, 3 | 14,238. Because 9 | 18, 9 | 14,238. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4-7 The store manager has an invoice for 72 calculators. The first and last digits on the receipt are illegible. The manager can read \$■67.9■ What are the missing digits, and what is the cost of each calculator? Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example (continued) Let the missing digits be represented by x and y, so that the number is x67.9y dollars, or x679y cents. Because 72 calculators were sold, the amount must be divisible by 72. Because 72 = 8 · 9, the amount is divisible by both 8 and 9. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example (continued) For the number on the invoice to be divisible by 8, the three-digit number 79y must be divisible by 8. Only 792 is divisible by 8, so y = 2, and the last digit on the invoice is 2. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example (continued) Because the number on the invoice must be divisible by 9, we know that 9 must divide x , or x + 24. Since 3 is the only single digit that will make x + 24 divisible by 9, x = 3. The number on the invoice must be \$ The calculators cost \$ ÷ 72 = \$5.11, each. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Divisibility Tests A whole number is divisible by 11 if and only if the sum of the digits in the places that are even powers of 10 minus the sum of the digits in the places that are odd powers of 10 is divisible by 11. An whole number is divisible by 6 if and only if the whole number is divisible by both 2 and 3. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4-8 The number 57,729,364,583 has too many digits for most calculator displays. Determine whether it is divisible by each of the following: a. 2 No b. 3 No c. 5 No d. 6 No e. 8 No f. 9 No g. 10 No h. 11 Yes Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

4-2 Prime and Composite Numbers
Prime Factorization Number of Divisors Determining if a Number is Prime More About Primes Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Prime and Composite Numbers
The following rectangles represent the number 18. 1 18 6 9 2 3 The number 18 has 6 positive divisors: 1, 2, 3, 6, 9 and 18. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Prime and Composite Numbers
Number of Positive Divisors Below each number listed across the top, we identify numbers less than or equal to 37 that have that number of positive divisors. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Prime Numbers Number of Positive Divisors
These numbers have exactly 2 positive divisors, 1 and themselves. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Composite Numbers Number of Positive Divisors
These numbers have at least one factor other than 1 and themselves. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Prime and Composite Numbers
Number of Positive Divisors The number 1 has only one positive factor – it is neither prime nor composite. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Definition Prime number Any positive integer with exactly two distinct, positive divisors Composite number Any whole number greater than 1 that has a positive factor other than 1 and itself Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4-9 Show that the following numbers are composite. a. 1564 Since 2 | 4, 1564 is divisible by 2 and is composite. b. 2781 Since 3 | ( ), 2781 is divisible by 3 and is composite. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example (continued) c. 1001 Since 11 | [(1 + 0) − (0 + 1)], 1001 is divisible by 11 and is composite. d. 3 · 5 · 7 · 11 · The product of odd numbers is odd, so 3 · 5 · 7 · 11 · 13 is odd. When 1 is added to an odd number, the sum is even. All even numbers are divisible by 2 and all even numbers, except 2, are composite. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Prime Factorization Students continue to develop their understanding of multiplication and division and the structure of numbers by determining if a counting number greater than 1 is a prime, and if it is not, by factoring it into a product of primes. NCTM grade 7 Curriculum Focal Points, p. 19 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Prime Factorization Composite numbers can be expressed as products of two or more whole numbers greater than 1. Each expression of a number as a product of factors is a factorization. A factorization containing only prime numbers is a prime factorization. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Fundamental Theorem of Arithmetic (Unique Factorization Theorem)
Each composite number can be written as a product of primes in one and only one way except for the order of the prime factors in the product. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Prime Factorization To find the prime factorization of a composite number, rewrite the number as a product of two smaller natural numbers. If these smaller numbers are both prime, you are finished. If either is not prime, then rewrite it as the product of smaller natural numbers. Continue until all the factors are prime. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Prime Factorization 84 495 4 21 5 99 11 9 2 2 3 7 3 3 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Prime Factorization The two trees produce the same prime factorization, except for the order in which the primes appear in the products. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Prime Factorization We can also determine the prime factorization by dividing with the least prime, 2, if possible. If not, we try the next larger prime as a divisor. Once we find a prime that divides the number, we continue by finding smallest prime that divides that quotient, etc. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Number of Divisors How many positive divisors does 24 have? We are not asking how many prime divisors, just the number of divisors – any divisors. 1, 2, 3, 4, 6, 8, 12, 24 Since 1 is a divisor of 24, then 24/1 = 24 is a divisor of 24. Since 2 is a divisor of 24, then 24/2 = 12 is a divisor of 24. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Number of Divisors 1, 2, 3, 4, 6, 8, 12, 24 Since 3 is a divisor of 24, then 24/3 = 8 is a divisor of 24. Since 4 is a divisor of 24, then 24/4 = 6 is a divisor of 24. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Number of Divisors Another way to think of the number of positive divisors of 24 is to consider the prime factorization 23 = 8 has four divisors. 3 has two divisors. Using the Fundamental Counting Principle, there are 4 × 2 = 8 divisors of 24. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Number of Divisors If p and q are different primes, m and n are whole number then pnqm has (n + 1)(m + 1) positive divisors. In general, if p1, p2, …, pk are primes, and n1, n2, …, nk are whole numbers, then has positive divisors. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4-10a Find the number of positive divisors of 1,000,000. The prime factorization of 1,000,000 is 26 has = 7 divisors, and 56 has = 7 divisors. has (7)(7) = 49 divisors. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4-10b Find the number of positive divisors of The prime factorization of is 210 has = 11 divisors, 310 has = 11 divisors, 510 has = 11 divisors, and 710 has = 11 divisors. has 114 = 14,641 divisors. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Determining if a Number is Prime
To determine if a number is prime, you must check only divisibility by prime numbers less than the given number. For example, to determine if 97 is prime, we must try dividing 97 by the prime numbers: 2, 3, 5, and so on as long as the prime is less than 97. If none of these prime numbers divide 97, then 97 is prime. Upon checking, we determine that 2, 3, 5, 7 do not divide 97. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Determining if a Number is Prime
Assume that p is a prime greater than 7 and p | 97. Then 97/p also divides 97. Because p ≥ 11, then 97/p must be less than 10 and hence cannot divide 97. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Determining if a Number is Prime
If d is a divisor of n, then is also a divisor of n. If n is composite, then n has a prime factor p such that p2 ≤ n. If n is a whole number greater than 1 and not divisible by any prime p, such that p2 ≤ n, then n is prime. Note: Because p2 ≤ n implies that it is enough to check if any prime less than or equal to is a divisor of n. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4-11a Is 397 composite or prime? The possible primes p such that p2 ≤ 397 are 2, 3, 5, 7, 11, 13, 17, and 19. Because none of the primes 2, 3, 5, 7, 11, 13, 17, and 19 divide 397,397 is prime. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 4-11b Is 91 composite or prime? The possible primes p such that p2 ≤ 91 are 2, 3, 5, and 7. Because 91 is divisible by 7, it is composite. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Sieve of Eratosthenes One way to find all the primes less than a given number is to use the Sieve of Eratosthenes. If all the natural numbers greater than 1 are considered (or placed in the sieve), the numbers that are not prime are methodically crossed out (or drop through the holes of the sieve). The remaining numbers are prime. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Sieve of Eratosthenes Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

4-3 Greatest Common Divisor and Least Common Multiple
Methods to Find the Greatest Common Divisor Methods to Find the Least Common Multiple

Greatest Common Divisor
Two bands are to be combined to march in a parade. A 24-member band will march behind a 30-member band. The combined bands must have the same number of columns. Each column must be the same size. What is the greatest number of columns in which they can march?

Greatest Common Divisor
The bands could each march in 2 columns, and we would have the same number of columns, but this does not satisfy the condition of having the greatest number of columns. The number of columns must divide both 24 and 30. Numbers that divide both 24 and 30 are 1, 2, 3, and 6. The greatest of these numbers is 6.

Greatest Common Divisor
The first band would have 6 columns with 4 members in each column, and the second band would have 6 columns with 5 members in each column.

Definition The greatest common divisor (GCD) or the greatest common factor (GCF) of two whole numbers a and b not both 0 is the greatest whole number that divides both a and b.

Colored Rods Method Find the GCD of 6 and 8 using the 6 rod and the 8 rod.

Colored Rods Method Find the longest rod such that we can use multiples of that rod to build both the 6 rod and the 8 rod. The 2 rods can be used to build both the 6 and 8 rods.

Colored Rods Method The 3 rods can be used to build the 6 rod but not the 8 rod. The 4 rods can be used to build the 8 rod but not the 6 rod. The 5 rods can be used to build neither. The 6 rods cannot be used to build the 8 rod. Therefore, GCD(6, 8) = 2.

The Intersection-of-Sets Method
List all members of the set of whole number divisors of the two numbers, then find the set of all common divisors, and, finally, pick the greatest element in that set.

The Intersection-of-Sets Method
To find the GCD of 20 and 32, denote the sets of divisors of 20 and 32 by D20 and D32, respectively. Because the greatest number in the set of common positive divisors is 4, GCD(20, 32) = 4.

The Prime Factorization Method
To find the GCD of two or more non-zero whole numbers, first find the prime factorizations of the given numbers and then identify each common prime factor of the given numbers. The GCD is the product of the common factors, each raised to the lowest power of that prime that occurs in any of the prime factorizations. Numbers, such as 4 and 9, whose GCD is 1 are relatively prime.

Example 4-12 Find each of the following: a. GCD(108, 72) b. GCD(0, 13)
Because 13 | 0 and 13 | 13, GCD(0, 13) = 13.

Example (continued) c. GCD(x, y) if x = 23 · 72 · 11 · 13 and y = 2 · 73 · 13 · 17 GCD(x, y) = 2 · 72 · 13 = 1274 d. GCD(x, y, z) if x = 23 · 72 · 11 · 13, y = 2 · 73 · 13 · 17, and z = 22 · 7 GCD(x, y, z) = 2 · 7 = 14 e. GCD(x, y) if x = 54 · 1310 and y = 310 · 1120 Because x and y have no common prime factors, GCD(x, y) = 1.

Calculator Method Calculators with a key can be used to find the GCD of two numbers. Simp Find GCD(120, 180) by pressing the keys: 1 Simp 2 / 8 = to obtain the display N/D→n/d 60/90

Calculator Method By pressing the key, we see on the display as a common divisor of 120 and 180. x y 2 By pressing the key again and pressing we see 2 again as a factor. Simp = x y Repeat the process to see that 3 and 5 are other common factors. GCD(120, 180) is the product of the common prime factors 2 · 2 · 3 · 5, or 60.

Euclidean Algorithm Method
If a and b are any whole numbers greater than 0 and a ≥ b, then GCD(a, b) = GCD(r, b), where r is the remainder when a is divided by b. Finding the GCD of two numbers by repeatedly using the theorem above until the remainder is 0 is called the Euclidean algorithm.

Euclidean Algorithm Method

Example 4-13 Use the Euclidean algorithm to find GCD(10764, 2300).
GCD(10764, 2300) = GCD(2300, 1564) GCD(2300, 1564) = GCD(1564, 736)

Example 4-13 (continued) GCD(10764,2300) = GCD(92, 0) = 92

Euclidean Algorithm Method
A calculator with the integer division feature can also be used to perform the Euclidean algorithm. To find GCD(10764, 2300), proceed as follows: The last number we divided by when we obtained the 0 remainder is 92, so GCD(10764, 2300) = 92.

Example 4-14a Find GCD(134791, 6341, 6339). Any common divisor of three numbers is also a common divisor of any two of them. The GCD of three numbers cannot be greater than the GCD of any two of the numbers. GCD(6341, 6339) = GCD(6341 − 6339, 6339) = GCD(2, 6339) = 1 GCD(134791, 6341, 6339) cannot be greater than 1, so it must equal 1.

Example 4-14b Find the GCD of any two consecutive whole numbers.
GCD(n, n + 1) = GCD(n + 1, n) = GCD(n + 1 − n, n) = GCD(1, n) = 1 The GCD of any two consecutive whole numbers is 1.

Least Common Multiple Hot dogs are usually sold 10 to a package, while hot dog buns are usually sold 8 to a package. What is the least number of packages of each you must buy so that there is an equal number of hot dogs and buns? The number of hot dogs is a multiple of 10, while the number of buns is a multiple of 8. The number of hot dogs matches the number of buns whenever 10 and 8 have multiples in common.

Least Common Multiple This occurs at 40, 80, 120…
The least of these multiples is 40. So we will have the same number of hot dogs and buns by buying 4 packages of hot dogs and 5 packages of buns.

Definition Least Common Multiple (LCM)
The least common multiple (LCM) of two non-zero whole numbers a and b is the least non-zero whole number that is simultaneously a multiple of a and a multiple of b.

Number-Line Method Find LCM(3, 4).
Beginning at 0, the arrows do not coincide until the point 12 on the number line. Thus, 12 is LCM(3, 4).

Colored Rods Method Find LCM(3, 4) using the 3 rod and the 4 rod.

Colored Rods Method Build trains of 3 rods and 4 rods until they are the same length. The LCM is the common length of the train. LCM(3, 4) = 12

The Intersection-of-Sets Method
List all members of the set of positive multiples of the two integers, then find the set of all common multiples, and, finally, pick the least element in that set.

The Intersection-of-Sets Method
To find the LCM of 8 and 12, denote the sets of positive multiple of 8 and 12 by M8 and M12, respectively. Because the least number in the set of common positive multiples is 24, LCM(8, 12) = 24.

The Prime Factorization Method
To find the LCM of two non-zero whole numbers, first find the prime factorization of each number. Then take each of the primes that are factors of either of the given numbers. The LCM is the product of these primes, each raised to the greatest power of the prime that occurs in either of the prime factorizations.

Example 4-15 Find LCM(2520, 10530).

GCD-LCM Product Method
For any two natural numbers a and b, GCD(a, b) · LCM(a, b) = ab.

Example 4-16 Find LCM(731, 952). Applying the Euclidean Algorithm, we can determine that GCD(731, 952) = 17. 17 · LCM(731, 952) = 731 · 952 LCM(731, 952) = = 40,936

Division-by-Primes Method
To find LCM(12, 75, 120), start with the least prime that divides at least one of the given numbers. Divide as follows: Because 2 does not divide 75, simply bring down the 75. To obtain the LCM using this procedure, continue the division process until the row of answers consists of relatively prime numbers as shown next.

Division-by-Primes Method