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

Chapter

Number Theory

44

Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

4-1 Divisibility

Divisibility

Divisibility Rules


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.


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.


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.


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 Trued.

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


Example 4-1 (continued)

h. True

g. 0 | 0 False


3| (5 7 9 11 1)

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.


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


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

Falsec.


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 = 1700 + 34. We know that 17 | 1700 because 17 | 17 and 17 divides any multiple of 17.

Divisibility Rules

Furthermore, 17 | 34; therefore, we conclude that 17 | 1734.


Another method to check for divisibility is to use the integer division button on a calculator.

Divisibility Rules

INT ÷

Press the following sequence of buttons:

INT ÷ 1 7 3 4 1 7 =

to obtain the display 102 0.Q R


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.

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.

Divisibility Tests


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.


Example 4-4b

Determine whether 83,026 is divisible by 2, 4, and 8.

2 | 83,026 because 2 | 6.


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.

Divisibility Tests


Example 4-6a

Determine whether 1002 is divisible by 3 and 9.

Because 1 + 0 + 0 + 2 = 3 and 3 | 3, 3 | 1002.

Because


Example 4-6b

Determine whether 14,238 is divisible by 3 and 9.

Because 1 + 4 + 2 + 3 + 8 = 18 and 3 | 18, 3 | 14,238.

Because 9 | 18, 9 | 14,238.


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?



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.



Only 792 is divisible by 8, so y = 2, and the last digit on the invoice is 2.

For the number on the invoice to be divisible by 8, the three-digit number 79y must be divisible by 8.



Since 3 is the only single digit that will make x + 24 divisible by 9, x = 3.

Because the number on the invoice must be divisible by 9, we know that 9 must dividex + 6 + 7 + 9 + 2, or x + 24.

The number on the invoice must be $367.92.

The calculators cost $367.92 ÷ 72 = $5.11, each.


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.

Divisibility Tests


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 Nod. 6No

e. 8 Nof. 9No

g. 10 Yesh. 11No


4-2 Prime and Composite Numbers

Prime and Composite Numbers

Prime Factorization

Number of Divisors

Determining if a Number is Prime

More About Primes


The following rectangles represent the number 18.


118

2

9

3

6

The number 18 has 6 positive divisors: 1, 2, 3, 6, 9 and 18.



Below each number listed across the top, we identify numbers less than or equal to 37 that have that number of positive divisors.

Number of Positive Divisors


Prime NumbersNumber of Positive Divisors

These numbers have exactly 2

positive divisors, 1 and

themselves.


These numbers have at least

one factor other than 1

and themselves.

Composite NumbersNumber of Positive Divisors


Number of Positive DivisorsThe number 1 has only one

positive factor – it is neither prime nor

composite.



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


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 | (2 + 7 + 8 + 1), 2781 is divisible by 3 and is composite.



c. 1001

Since 11 | [(1 + 0) − (0 + 1)], 1001 is divisible by 11 and is composite.

d. 3 · 5 · 7 · 11 · 13 + 1

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.


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


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.


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.


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.

Prime Factorization


Prime Factorization

84

4

3 72 2

21

495

99

9

33

11

5


Prime Factorization

The two trees produce the same prime factorization, except for the order in which the primes appear in the products.


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.

Prime Factorization


Prime Factorization


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.

Since 1 is a divisor of 24, then 24/1 = 24 is a divisor of 24.


1, 2, 3, 4, 6, 8, 12, 24


Number of Divisors



1, 2, 3, 4, 6, 8, 12, 24


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.


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.


Example 4-10a

Find the number of positive divisors of 1,000,000.

The prime factorization of 1,000,000 is

26 has 6 + 1 = 7 divisors, and 56 has 6 + 1 = 7 divisors.

has (7)(7) = 49 divisors.


Example 4-10b

Find the number of positive divisors of 21010.

The prime factorization of 21010 is

210 has 10 + 1 = 11 divisors, 310 has 10 + 1 = 11 divisors, 510 has 10 + 1 = 11 divisors, and 710 has 10 + 1 = 11 divisors.

has 114 = 14,641 divisors.


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.




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.



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.


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.


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.


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.


Sieve of Eratosthenes


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?


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.


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.

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.

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.

The Prime Factorization Method

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.


c. GCD(x, y) if x = 23 · 72 · 11 · 13 and y = 2 · 73 · 13 · 17

d. GCD(x, y, z) if x = 23 · 72 · 11 · 13, y = 2 · 73 · 13 · 17, and z = 22 · 7

GCD(x, y) = 2 · 72 · 13 = 1274

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 Simp2 0 / 1 8 0 =

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.

2x y

By pressing the key again and pressing

we see 2 again as a factor.Simp =

x y

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.

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.

Euclidean Algorithm Method

Finding the GCD of two numbers by repeatedly using the theorem above until the remainder is 0 is called the Euclidean algorithm.


Example 4-13

Use the Euclidean algorithm to find GCD(10764, 2300).

GCD(10764, 2300) = GCD(2300, 1564)

GCD(2300, 1564) = GCD(1564, 736)


GCD(2300, 1564) = GCD(1564, 736)

GCD(1564, 736) = GCD(736, 92)

GCD(736, 92) = GCD(92, 0) = 92

GCD(10764,2300) = GCD(92, 0) = 92


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.

Find LCM(3, 4).

Number-Line Method

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

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

Colored Rods Method

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

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.


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.

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.

The Prime Factorization Method

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.

Applying the Euclidean Algorithm, we can determine that GCD(731, 952) = 17.

Example 4-16

Find LCM(731, 952).

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

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