Chapter 5 Properties of Whole Numbers

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Chapter 5Properties of Whole Numbers
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Some preliminary definitions

• If we have whole numbers a, b, and c such that
• a b c
• then we say that
• a divides c or
• a is a factor of c or
• a is a divisor of c and
• c is a multiple of a or
• c is divisible by a
• (We can always interchange a with b.)

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Properties of Divisibility
Suppose that a, m, n, k are whole numbers where a
? 0.

• If a divides m and a divides n, then a divides (m
  n).
• If a divides m but a does not divide n, then a
  does not divide (m n).
• If a divides both m and n, and m n, then a
  divides (m n).
• If a divides m but a does not divide n, and m
  n, then a does not divide (m n).
• If a divides m, then a divides km.

Transitive Property of Divisibility If a, b, c
are non zero whole numbers such that a divides
b and b divides c, then a divides c.
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• Important remarks
• 0 is divisible by any non-zero number,
• No number is divisible by 0.

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Different types of whole numbers

• Prime numbersA number is prime if it is gt 1 and
  it is only divisible by 1 and itself.e.g. 2, 3,
  5, 7, 11,
• Composite numbersA number is composite if it can
  be written as the product of two smaller whole
  numbers.e.g. 4 (22), 6 (23), 10 (25),
• IdentitiesThe numbers 0 and 1 are not prime nor
  composite, they are just identities for
  respective operations.

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Prime numbers play an important role within
the whole numbers because they are the primary
components. More precisely, we have the
following Theorem Every whole number bigger than
1 is either a prime number or a product of prime
numbers. (in other words, the prime numbers can
generate all whole numbers bigger than 1 by
multiplication.)
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Factorization Trees According to the previous
theorem, if a number n is not prime, we must be
able to break it down to a product of prime
numbers. Here is how,
60
This tree is by no means unique, we can easily
make another one, such as
60
6
10
2
30
2
3
2
5
5
6
2
3
However, the collection of prime numbers we get
from the leaves of the tree is always the
same. In other words, 60 2235 no matter
how we factorize it.
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Unique Factorization Theorem Any whole number
bigger than 1 can be factorized into a product of
prime numbers in exactly one way if we list its
prime factors from small to large.
2233357
Example 3780
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Test for Primeness (i.e. a faster way to
determine whether a whole number is prime or not)
Theorem A whole number n ( gt1) is prime if it is
not divisible by any prime number ?
Example Is 179 a prime number? a. compute
which is approx. 13.379
b. all prime numbers ? 13.3 are 2, 3, 5, 7, 11,
13 c. Is 179 divisible by 2? No Is 179
divisible by 3? No Is 179 divisible by 5?
No Is 179 divisible by 7? No Is 179
divisible by 11? No Is 179 divisible by 13?
No Therefore 179 is prime. (Note this reduce
the work from checking 177 numbers to just
6 numbers!)
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Divisibility Tests
Test of divisibility for 2 A whole number n is
divisible by 2 if and only if its last digit is
divisible by 2 (i.e. even)
Test of divisibility for 5 A whole number n is
divisible by 5 if and only if its last digit is
divisible by 5 (i.e. 0 or 5)
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Divisibility Tests
Test of divisibility for 4 A whole number n
is divisible by 4 if and only if the number
represented by its last two digits is divisible
by 4. Examples 21732 is divisible by 4 because
32 is divisible by 4. Note that neither the
digit 3 nor 2 is divisible by 4. It is the number
3102 that is divisible by 4. Hence it is wrong
to say that the last two digits are divisible by
4. 44442 is not divisible by 4 because 42 is not.
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Divisibility Tests
Test of divisibility for 8 A whole number n
is divisible by 8 if and only if The number
represented by its last three digits is divisible
by 8. Examples 201432 is divisible by 8
because 432 is divisible by 8, 716510 is not
divisible by 8 because 510 is not divisible by 8

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Divisibility Tests
Test of divisibility for 3 A whole number n
is divisible by 3 if and only if the sum of its
digits is divisible by 3.
Example 71653092 is divisible by 3 because
7 1 6 5 3 0 9 2 33

Test of divisibility for 9 A whole number n
is divisible by 9 if and only if the sum of its
digits is divisible by 9.
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Divisibility Tests
Test of divisibility for 11 A whole number
n is divisible by 11 if and only if the
alternate sum of its digits is divisible by 11.
Examples Lets first look at all 2-digit
multiples of 11, 11, 22, 33, 44, 55, 66, 77, 88,
99 The difference of the two digits is 0, hence
divisible by 11.

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Divisibility Tests
Test of divisibility for 11 A whole number
n is divisible by 11 if and only if the
alternate sum of its digits is divisible by 11.
Examples 3-digit multiples of 11 121,
132, 154, 165, ,
957, 968, 979, 990. You should
see that 1st digit 2nd digit
3rd digit is divisible by 11.
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Divisibility Tests
Test of divisibility for 11 A whole number
n is divisible by 11 if and only if the
alternate sum of its digits is divisible by 11.
More examples 72952 is divisible by 11 because
7 2 9 5 2 11, which is divisible by
11. 56823 is not divisible by 11 because 5
6 8 2 3 8, which is not divisible by 11.
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Greatest Common Factor (GCF)
Definition The GCF of two whole numbers a and b
is the largest whole number c that can divide
into both a and b.

• Methods for finding GCF(a, b)
• Set intersection method depends on
  listing all factors of a and b.

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Greatest Common Factor (GCF)
Definition The GCF of two whole numbers a and b
is the largest whole number c that can divide
into both a and b.

• Methods for finding GCF(a, b)
• Set intersection method depends on
  listing all factors of a and b.example to find
  the GCF(12, 18)

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Greatest Common Factor (GCF)
Definition The GCF of two whole numbers a and b
is the largest whole number c that can divide
into both a and b.

• Methods for finding GCF(a, b)
• Set intersection method depends on
  listing all factors of a and b.example to find
  the GCF(12, 18)factors of 12 1, 2, 3, 4, 6,
  12.

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Greatest Common Factor (GCF)
Definition The GCF of two whole numbers a and b
is the largest whole number c that can divide
into both a and b.

• Methods for finding GCF(a, b)
• Set intersection method depends on
  listing all factors of a and b.example to find
  the GCF(12, 18)factors of 12 1, 2, 3, 4, 6,
  12.factors of 18 1, 2, 3, 6, 9, 18.

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Greatest Common Factor (GCF)
Definition The GCF of two whole numbers a and b
is the largest whole number c that can divide
into both a and b.

• Methods for finding GCF(a, b)
• Set intersection method depends on
  listing all factors of a and b.example to find
  the GCF(12, 18)factors of 12 1, 2, 3, 4, 6,
  12.factors of 18 1, 2, 3, 6, 9, 18.Clearly
  the red numbers are the common factors and 6 is
  the largest, ? GCF(12, 18) 6

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Greatest Common Factor (GCF)

• Methods for finding GCF(a, b)
• (method I is straight forward and easy to learn
  but slow for larger numbers)
• Factorization method depends on the
  complete factorization of a and b.example to
  find the GCF(24, 180)

24 2223, 180 22335 and
by the Greedy algorithm, we pick all the prime
factors common to both numbers (including
multiplicities) ? GCF(24, 180) 223
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• (method II is still difficult to use if the
  numbers are too large and difficult
• to factorize.)
• Euclidean Algorithm depends on reducing
  large numbers to smaller ones.Theory if a gt b,
  then GCF(a, b) GCF(a - b, b)in other words,
  we can reduce the larger number by subtracting it
  with the smaller one, without changing the GCF.

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• (method II is still difficult to use if the
  numbers are too large and difficult
• to factorize.)
• Euclidean Algorithm depends on reducing
  large numbers to smaller ones.Theory if a gt b,
  then GCF(a, b) GCF(a - b, b)in other words,
  we can reduce the larger number by subtracting it
  with the smaller one, without changing the GCF.

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• Euclidean Algorithm depends on reducing
  large numbers to smaller ones.Theory if a gt b,
  then GCF(a, b) GCF(a - b, b)in other words,
  we can reduce the larger number by subtracting it
  with the smaller one, without changing the GCF.

Example GCF(481, 296)
GCF(481 296, 296) GCF(185, 296) GCF(185,
296 185) GCF(185, 111) GCF(74, 111)
GCF(74, 37) GCF(37, 37) 37
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Lowest Common Multiple (LCM)
Definition The LCM of two whole numbers a and b
is the smallest whole number c that is divisible
by both a and b.

• Methods for finding LCM(a, b)
• Set intersection method depends on
  listing small multiples of a and b.example to
  find the LCM(15, 24)multiples of 15 multiples
  of 24

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Lowest Common Multiple (LCM)
Definition The LCM of two whole numbers a and b
is the smallest whole number c that is divisible
by both a and b.

• Methods for finding LCM(a, b)
• Set intersection method depends on
  listing small multiples of a and b.example to
  find the LCM(15, 24)multiples of 15 multiples
  of 24

15, 30,
45, 60,
75,
90, 105,
24, 48,
72,
96,
120,
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Lowest Common Multiple (LCM)
Definition The LCM of two whole numbers a and b
is the smallest whole number c that is divisible
by both a and b.

• Methods for finding LCM(a, b)
• Set intersection method depends on
  listing small multiples of a and b.example to
  find the LCM(15, 24)multiples of 15 multiples
  of 24

15, 30,
45, 60,
75,
90, 105,
120.
24, 48,
72,
96,
120,
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Lowest Common Multiple (LCM)
Definition The LCM of two whole numbers a and b
is the smallest whole number c that is divisible
by both a and b.

• Methods for finding LCM(a, b)
• Set intersection method depends on
  listing small multiples of a and b.example to
  find the LCM(15, 24)multiples of 15 multiples
  of 24

15, 30,
45, 60,
75,
90, 105,
24, 48,
72,
96,
Therefore LCM(15, 24) 120.
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Lowest Common Multiple (LCM)

• Methods for finding LCM(a, b)
• Factorization method a thrifty
  algorithmexample to find the LCM(875,
  1225)875 5557 1225
  5577Analogy Your math teacher requires you
  to bring 3 pencils and 1 highlighter to class,
  your History teacher requires you to bring 2
  pencils and 2 highlighters. What is the minimum
  set of pens you need to bring if you have both
  classes in one day?

Ans 3 pencils and 2 highlighters.
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? LCM(875, 1225)
55577
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Lowest Common Multiple (LCM)

• Methods for finding LCM(a, b)
• Factorization method Another example to
  find the LCM(60, 126)60 2235
  126 2337

Answer
223357
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Lowest Common Multiple (LCM)

• Methods for finding LCM(a, b)
• Mechanical method using a formula

Example Find the LCM of 481 and 296. Answer
we have calculated before that GCF(481, 296)
37, therefore LCM(481,
296) (481296) 37 3848
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Lowest Common Multiple (LCM)
Example Your neighbor has two dogs - one named
Sparky and one named Skippy. They both like to
bark and both have a pattern. Sparky will bark
for a while every 15 minutes once started, and
Skippy does the same every 18 minutes. One night
they both started barking at 9pm because a
cat passed by. When did they bark together again
that night?