Transcript Chapter 11

Reconceptualizing
Mathematics
Part 1: Reasoning About Numbers and Quantities
Judith Sowder, Larry Sowder, Susan Nickerson
CHAPTER 11 – NUMBER THEORY
1
© 2010 by W. H. Freeman and Company. All rights reserved.
Recall the discussion of factors from Chapter 3. Since 5 
6 = 30, and 15  2 = 30, each of the numbers 2, 5, 6, and
15 are factors of 30 (and of course there are more).
11-2
Discussion
11-3
ACTIVITY
On a sheet of paper, create the array of numbers shown below:
11-4
continued….
1. a) Cross out 1 in the array.
b) The number 2 is prime. Circle 2 in the array. Cross out all the
larger multiples of 2 in the array.
c) Do the same now for 3, 5, and 7.
d) What is circled next? Does this procedure ever end?
e) Circle 11 in the array. Cross out all the larger multiples of 11.
f) Circle all the numbers not yet crossed out.
g) Are the numbers circled prime? Can you surmise why it turned
out this way?
2. a) If the array were extended, what column would 1000 be in?
How about 1,000,000?
b) 210 = 1024. What column would this be in?
c) 211 = 2048. What column would this be in?
3. Find columns in the array for which the following is true: If two
numbers in the column are multiplied, the product is also in that
column. What is the name of this property?
11-5
11-6
Example:
11-7
ACTIVITY
11-8
ACTIVITY
In a certain school there are 100 lockers lining a hallway. All are closed.
Suppose 100 students walk down the hallway, in file, and the first student
opens every locker. The second student comes behind the first and closes
every second locker (beginning with number two). The third student
changes the position of every third locker (if open, it gets closed, and vice
versa). The fourth student changes the position of every fourth locker, and
so on, until the 100th student changes the position of every 100th locker.
After this procession, which lockers are open? Why? At the end of the
procession, how many times did lockers nine and ten get changed?
11-9
11.1
11-10
The number 6 can be written as a product of prime
numbers: 2  3. The number 18 can be written as a
product of three primes: 2  3  3. Can other composite
numbers be written as a product of primes? These
questions and others are explored next.
11-11
ACTIVITY
continued….
11-12
11-13
11-14
ACTIVITY
11-15
So, 2100 = 22  3  52  7
11-16
Activity
11-17
11-18
ACTIVITY
Return to the table we looked at in the first activity of
this section. Write each number using exponents where
possible. How can you determine the number of factors
of a composite number by knowing the exponents of its
prime factorization? Determine a rule for the number of
factors for any whole number other than zero.
Hint: For any prime number  to the nth power,  can
appear in a factor n + 1 ways where the exponents
are_______ (you finish the sentence—recall the rows of
factors we discussed for 72). Take note of the difference
between finding factors and finding prime factors.
11-19
11.2
11-20
It is a challenge to tackle large numbers to see whether
they are prime. It was newsworthy in 1995 when a team
of 600 volunteers with computers determined that a
number with 129 digits was prime. It took them eight
months to determine this.
11-21
Is 495,687,115 a prime? Is 2,298,543,316 a prime? It is
likely you saw immediately that five is a factor of the
first number and that two is a factor of the second.
Hence neither is prime.
From looking at this it is easy to see that divisibility tests
for 2, 5, and 10 are quite straightforward. A divisibility
test tells one whether a number is a factor of a given
number without having to complete the entire division
process.
11-22
11-23
You may recall the following two properties:
1) If k is a factor of both m and n, then k is a factor
of m + n.
2) If k is a factor of m, but not of n, then k is not
a factor of m + n.
11-24
Determining a divisibility test for three requires more
thought. The last digit is not going to be able to help us.
For example, 26 is not divisible by three. And while it’s
true that the prime factorization could tell us if a
number is divisible by three (why?), that can be a
lengthy process.
11-25
11-26
11-27
An interesting and useful fact about dividing a number
by nine is that the remainder is always the sum of digits
of the number. For example, 215 ÷ 9 = 23 with a
remainder of eight, and 2 + 1 + 5 = 8.
11-28
ACTIVITY
Use the reasoning from the divisibility test for three,
along with the reasoning for the remainder being the
sum of digits when dividing by nine, to derive a
divisibility test for 9.
11-29
11-30
11-31
11-32
ACTIVITY
11-33
We already know that four divides 100, 1000, 10000, and so on. So
consider:
11-34
ACTIVITY
Use the reasoning for the divisibility test for four to
construct a divisibility test for eight.
11-35
What about a divisibility test for six? Well, we already
know tests for two and three. So if a number passes
those two divisibility tests, then it must also be divisible
by six (why?).
The same could be said for a divisibility test for 12. Here
we could use the tests for four and three (how?).
11-36
Two numbers are relatively prime if they have no prime
factor in common.
Discussion
Are 12 and 6 relatively prime?
Are 15 and 6 relatively prime?
Are 25 and 6 relatively prime?
Are 7 and 11 relatively prime?
Are any two prime numbers relatively prime?
11-37
11-38
Divisibility tests can help us with prime factorizations of
large numbers as well. Consider 12,320. We can tell
right away that 2 and 5 divide 12,320, so we have:
2  5  1232
We know that 4 divides 1232, so:
2  5  4  308 = 2  5  2  2  308
But 4 also divides 308:
2  5  2  2  2  2  77
But it is clear enough that 77 is divisible by 11:
2  2  2  2  2  5  7  11
11-39
ACTIVITY
Use the method just explained to find the prime
factorization of 1224. Try it again for 4620.
11-40
11-41
DISCUSSION
11-42
One important conclusion that can be made from the
previous discussion is that you need to test only for
divisibility by primes when deciding whether a number is
prime. If, for example, 7 is not a factor, then neither will
14, 21, 28, etc. be factors.
11-43
DISCUSSION
11-44
11-45
11.3
11-46
The greatest common factor (GCF) of two numbers m
and n, sometimes also called the greatest common
divisor, is the largest number that divides both m and n.
In the illustration below, 6 is the GCF:
11-47
The least common multiple (LCM) of two numbers m
and n is the smallest number that is a multiple of m and
also a multiple of n. In the illustration below, 48 is the
LCM:
11-48
ACTIVITY
1. What is the GCF of 68 and 102?
2. What is the LCM of 68 and 102?
3. Simplify 68/102.
4. What is 5/68 + 14/102?
11-49
11-50
11-51
ACTIVITY
11-52
EXAMPLE
11-53
ACTIVITY
11-54
ACTIVITY
11-55
11.4
11-56
Once a number is represented as a product of prime
numbers, it is quite easy to find the factors and
multiples of the number. This method is possible
because the prime factorization is unique. As you know,
this is sometimes called the Fundamental Theorem of
Arithmetic and other times is referred to as the Unique
Factorization Theorem. Research has shown that
students have difficulty applying this theorem.
11-57
11-58
11-59
11.5
11-60
continued….
11-61
11-62