Download BB Chapter 4 - WordPress.com

Thinking Critically
4.1 Divisibility Of Natural Numbers
4.2 Tests for Divisibility
4.3 Greatest Common Divisors and Least
Common Multiples
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
4.1
Divisibility and Natural
Numbers
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-2
DEFINITION:
DIVIDES, FACTORS, DIVISOR, MULTIPLE
If a and b are whole numbers with b ≠ 0
and there is a whole number q such that
a = bq, we say that b divides a.
We also say that b is a factor of a or a
divisor of a and that a is a multiple of b.
If b divides a and b is less than a, it is
called a proper divisor of a.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-4
DEFINITION:
EVEN AND ODD WHOLE NUMBERS
A whole number a is even precisely when
it is divisible by 2.
a = 2k for a whole number k.
A whole number b that is not even is
called an odd number.
b = 2j + 1 for a whole number j.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-5
DEFINITION:
PRIMES, COMPOSITE NUMBERS, UNITS
A natural number that possesses exactly
two different factors, itself and 1, is called
a prime number.
A natural number that possesses more
than two different factors is called a
composite number.
The number 1 is called a unit; it is neither
prime nor composite.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-6
THE FUNDAMENTAL THEOREM OF
ARITHMETIC (Simple-Product Form)
Every natural number greater than 1 is a
prime or can be expressed as a product
of primes in one, and only one, way apart
from the order of the prime factors.
NOTE: This is why we do not think of 1
as either a prime or composite number, to
preserve uniqueness of the product.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-7
Example 4.3 Prime Factors of 600
Write 600 as a product of primes.
USING A FACTOR TREE
USING SHORT DIVISION
600 = 2 • 2 • 2 • 3 • 5 • 5
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-8
THE FUNDAMENTAL THEOREM OF
ARITHMETIC (Prime-Power Form)
Every natural number n greater than 1 is
a power of a prime or can be expressed
as a product of powers of different primes
in one, and only one, way apart from
order.
This representation is called the primepower representation of n.
600 = 2 • 2 • 2 • 3 • 5 • 5 = 23 • 31 • 52
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-9
PRIMES
The Number of Primes
There are infinitely many primes.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-12
4.2
Tests for Divisibility
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-10
DIVISIBILITY OF SUMS AND
DIFFERENCES
Let d, a, and b be natural numbers. Then if
d divides both a and b, then it also
divides their sum, a + b, and their
difference, a – b.
Example: 3 divides both 36 and 15, thus it
also divides 36 + 15 = 51 and 36 – 15 = 21.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-14
TESTS FOR DIVISIBILITY
By 2:
A natural number is divisible by 2 exactly
when its base ten units digit is 0, 2, 4, 6, or 8.
By 5:
A natural number is divisible by 5 exactly
when its base ten units digit is 0 or 5.
By 10:
A natural number is divisible by 10 exactly
when its base ten units digit is 0.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-15
TESTS FOR DIVISIBILITY
By 4:
A natural number is divisible by 4 when the
number represented by its last two digits is
divisible by 4.
By 8:
A natural number is divisible by 8 when the
number represented by its last three digits is
divisible by 8.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-16
TESTS FOR DIVISIBILITY
Is 81,164 divisible by 4 and by 8?
81,164 is divisible by 4 since 64
is divisible by 4.
81,164 is not divisible by 8
since 164 is not divisible by 8.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-17
TESTS FOR DIVISIBILITY
By 3:
A natural number is divisible by 3 if and only
if the sum of its digits is divisible by 3.
By 9:
A natural number is divisible by 9 if and only
if the sum of its digits is divisible by 9.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-18
TESTS FOR DIVISIBILITY
Is 81,165 divisible by 3 and by 9?
Sum of digits: 8 + 1 +1 + 6 + 5 = 21
81,164 is divisible by 3 since 21
is divisible by 3.
81,164 is not divisible by 9 since
21 is not divisible by 9.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-19
TESTS FOR DIVISIBILITY
By 11:
A natural number is divisible by 11 exactly
when the sum of its digits in the even and
odd positions have a difference that is
divisible by 11.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-20
Example 4.9 Test for divisibility by 11
Is 42,315,690 divisible by 11?
Even positions: 2 + 1 + 6 + 0 = 9
Odd positions: 4 + 3 + 5 + 9 = 21
Difference: 21 – 9 = 12 which is not
divisible by 11.
It follows that 42,315,690 is not
divisible by 11.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-21
4.3
Greatest Common Divisors
and Least Common Multiples
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-19
DEFINITION:
GREATEST COMMON DIVISOR
Let a and b be whole numbers not both 0.
The greatest natural number d that divides
both a and b is called their greatest
common divisor and we write
d  GCD(a, b ).
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-26
Example 4.12 Finding the GCD by
Intersection of Sets
Find the greatest common divisor of 24
and 27. (Rainbow)
List the sets of divisors of each number.
D24  {1, 2,3, 4, 6,8,12, 24}
D27  {1,3,9, 27}
Find the intersection of these sets.
D24  D27  {1,3}
GCD(24, 27)  3
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-27
FINDING THE GCD:
PRIME FACTORIZATION METHOD
Let a and b be natural numbers. Then the
GCD(a, b) is the product of the prime powers
in the prime-power factorizations of a and b
which have the smaller exponents (including
zero).
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-28
FINDING THE GCD:
PRIME FACTORIZATION METHOD
Compute the greatest common divisor of
m  2  3  5 and n  3  5  7.
2
3
4
Rewrite the numbers as follows:
m  2  3  5  7 and n  2  3  5  7
1
2
3
0
0
4
1
1
GCD(m, n)  20  32  51  70  9  5  45
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-29
DEFINITION:
LEAST COMMON MULTIPLE
Let a and b be natural numbers. The least
natural number m that is a multiple of both a
and b is called their least common multiple,
and we write
m  LCM(a, b).
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-32
Example 4.15 Finding a LCM by Set
Intersections
Find the least common multiple of 9 and 15.
List the sets of multiples of each number.
M 9  {9,18, 27,36, 45,54,63,72,81,90,...}
M15  {15,30, 45,60, }
Find the intersection of these sets.
M 9  M15  {45,90, }
LCM(9,15)  45
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-33
FINDING THE LCM:
PRIME FACTORIZATION METHOD
Let m and n be natural numbers. Then the
LCM(m,n) is the product of the prime powers
in the prime-power factorizations of m and n
that have the larger exponents.
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
Slide 4-34
Example Finding the LCM: Prime-Power
Method
Compute the least common multiple of
2
3
4
m  2250  2  3  5 and n  2835  3  5  7.
Rewrite the numbers as follows:
m  2  3  5  7 and n  2  3  5  7
1
2
3
0
0
4
1
1
LCM(m, n)  2  3  5  7  141,750
1
4
Copyright © 2012, 2009, and 2006, Pearson Education, Inc.
3
1
Slide 4-35