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Transcript
Chapter 5
Number Theory
5.1 Primes, composites, and tests for
divisibility
Primes and Composites
† Prime number (prime) – a counting
number with exactly two different
factors
† Composite number (composite) – a
counting number with more than two
factors
† Primes 2,3,5,7,11 only have
themselves and 1 as factors
Factor Trees
† Prime factorizaton – each tree takes
you to 60+2*2*3*5
Fundamental Theorem of
Arithmetic
† Each composite number can be
expressed as the product of primes in
exactly one way (except for the order
of the factors).
† Example: express 180 as the product
of primes
† 10*18 = 2*5*2*3*3 = 22*32*5
Divides
† Let a and b be any whole numbers
with a ≠ 0 . We say that a divides b,
and write a b if and only if there is a
whole number x such that ax=b. The
symbol a b means that a does not
divide b
Divides (cont)
† a divides b if and only if a is a factor
of b. When a divides b, we can also
say that a is a divisor of b, a is a
factor of b, b is a multiple of a, and
b is divisible by a.
† 3 12?
† 216 IS A MULTIPLE OF 6 ?
† 8 IS A DIVISOR OF 96 ?
Test for divisibility by 2,5, and 10
† A number is divisible by 2 if and only
if its ones digit is 0,2,4,6, or 8.
† A number is divisible by 5 if and only
if its ones digit is 0 or 5
† A number is divisible by 10 if and
only if its ones digit is 0
Theorem:
† Let a, m, n, and k be whole numbers
where a ≠ 0
a) If a m and a n then a (m+n)
b) If a m and a n then a (m-n) for
c) If a m, then a km.
3 (9+12)
m≥n
Tests for divisibility by 4 and 8
† A number is divisible by 4 if and only
if the number represents by its last
two digits is divisible by 4
† A number is divisible by 8 if and only
if the number represented by its last
three digits is divisible by 8
† 4 1432 ?
† 8 98,765,432 ?
Tests for divisibility by 3 and 9
† A number is divisible by 3 if and only
if the sum of its digits is divisible by 3
† A number is divisible by 9 if and only
if the sum of its digits is divisible by 9
† 3 12,345 ?
† 9 6543 ?
Test for Divisibility by 11
† A number is divisible by 11 if and
only if 11 divides the difference of the
sum of the digits whose place values
are odd powers of 10 and the sum of
the digits whose place values are
even powers of 10
† 11 5346
† 11 909,381
Test for Divisibility by 6
† A number is divisible by 6 if and only
if both of the tests for divisibility by 2
and 3 hold.
† A number is divisible by 2 if and only
if its ones digit is 0,2,4,6, or 8.
† A number is divisible by 3 if and only
if the sum of its digits is divisible by 3
Theorem:
† A number is divisible by the product,
ab, of two nonzero whole numbers a
and b if it is divisible by both a and b,
and a and b have only the number 1
as a common factor.
Prime Factor Test
† To test for prime factors of a number
n, one need only search for prime
2
factors p of n, where p ≤ n( or _ p ≤ n )
† Determine whether the following are
prime
„ 299
„ 401
Section 5.2
† Counting Factors
† Greatest Common Factor
† Least Common Multiple
Theorem
† Suppose that a counting number n is
expressed as a product of distinct primes
with their respective exponents,
† say n = ( p1n )( p2n )...( pmn )
Then the number of factors of n is the
product (n1 + 1) * (n2 + 1)... (nm + 1)
1
2
m
(The number of factors depends on the
exponents NOT Prime factors) .
Greatest Common Factor
† The greatest common factor
(GCF) of two (or more) nonzero
whole numbers is the largest
whole number that is a factor of
both (all) of the numbers. The
GCF of a and b is written
GCF(a,b). .
GCF – Set intersection method
1.
Find all factors
† GCF of 24 and 36 24=23*3 then 4*2=8 factors and
36 = 22*32 then 3*3=9 factors of 36
† Factors for 24 = ?
†
†
Factors for 36 = ?
†
2.
{1,2,3,4,6,8,12,24}
{1,2,3,4,6,9,12,18,36}
Find all common factors by taking the intersection of
the two sets from step 1
† {1,2,3,4,6,8,12,24} I {1,2,3,4,6,9,12,18,36} = {1,2,3,4,6,12}
3. Find the largest number in the set of common factors
in step 2
† Therefore 12 is the GCF of 24 and 36 .
GCF- Prime Factorization Method
1. Express the numbers in their prime factor
exponential form
† 24=23*3 and 36=22*32
2. The GCF will be the number 2m3n where m
is the smaller of the exponents of the 2s
and n is the smaller of the exponents of
the 3s
† For 23*3 and 22*32 m is the smaller of 3 and 2
and n is the smaller of 1 and 2
† Then GCF is 22*31=12 .
Theorem
† If a and b are whole numbers with
a ≥ b, then GCF(a,b) = GCF(a – b,b)
„ Find the GCF(546,390)
† GCF(546,390) = GCF(546-390,390)
†
= GCF(156,390)
†
= GCF(390-156,156)
†
= GCF(234,156)
†
=GCF(234-156,156)
†
= GCF(78,156)
†
= GCF (156-78,78)
†
= GCF (78,78) .
Euclidean Algorithm
† If a and b are whole numbers, with
a ≥ b and a ≥ bq + r where r < b
then GCF(a,b)=GCF(r,b)
† To find the GCF of any two numbers
this theorem can be applied
repeatedly until a remainder of zero
is obtained. The final divisor that
leads to the zero remainder is the
GCF .
Euclidean Algorithm
4 R 72
840 3432
11R 48
72 840
1R 24
48 72
2 R0
24 48
Least Common Multiple
† The least common multiple
(LCM) of two (or more) nonzero
whole numbers is the smallest
nonzero whole number that is a
multiple of each (all) of the
numbers. Written as LCM(a,b)
„ Useful when adding or subtracting
fractions
LCM – Set Intersection Method
1. List the first several nonzero multiples of
24 and 36
† Multiples of 24 {24,48,72,96,120,144…}
† Multiples of 36 {36,72,108,144,…}
2. Find the first several common multiples of
24 and 36 by taking the intersection of the
sets
† {24,48,72,96,120,144,...} I {36,72,108,144,... = {72,144,...}
3. Find the smallest number in the union from
step 2 = LCM (72)
LCM – Prime Factorization method
1. Express the numbers in their prime
factor exponential form
† 24=23*3 and 36=22*32
2. The LCM will be the number 2r3s
where r is the larger of the
exponents of the twos and s is the
larger of the exponents of the threes
† Then r is 3 and s is 2 then the LCM is
23*32=72
LCM – Build-up Method
1. Express the numbers in their prime factor
exponential form
† 24=23*3 and 36=22*32
2. Select the prime factorization of one of the
numbers and build the LCM by taking the
highest exponent of each factor
† Begin with 24=23*3 and compare the 2’s factor
with 36=22*32 and take the largest 23
† Then take the prime factor 3 and compare with
the 3’s factor of 32 and take the largest – 32
† Build up to 23*32* = LCM=72