Download Positive Integers

Introduction
Prepared by Sa’diyya Hendrickson
Name:
Date:
Package Summary
• Exponent Form and Basic Properties
• Order of Operations
• Using Divisibility Rules
• Finding Factors and Common Factors
• Primes, Prime Decomposition and Factor Trees
• Greatest Common Factors
• Least Common Multiples
• Let’s Play! (Exercises)
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Some Properties
1. Positive Integers (Z+ ):
The set of positive integers is the set Z+ = {1, 2, 3, 4, 5, . . . . . .}.
2. The Exponential Form an :
For some number a and some positive integer n, an simply means to multiply a
exactly n times (i.e. repeated multiplication).
Using algebra, we have:
an = a
· · · · a · a} (This is the expanded form)
| · a · · {z
n times
Here, the base a has been “raised to the exponent/power of n.” When dealing
with the exponents 2 and 3, it is common to say “a squared ” or “a cubed.”
e.g. “2 cubed” means to multiply
exactly three times. i.e. 23 = 2 · 2 · 2 =
.
Let a, b, and c be positive integers.
1. Commutative Property of addition: a + b = b + a
i.e. The order that we add doesn’t matter! e.g. 7 + 2 = 2 + 7
2. Commutative Property of multiplication: a · b = b · a
i.e. The order that we multiply doesn’t matter! e.g. 7 · 2 = 2 · 7
3. Distributive Property: a(b+c) = a(b)+a(c). [The right side is in expanded form.]
i.e. the a distributes over the sum inside of the brackets. e.g. 2(3 + 4) = 2(3) + 2(4)
**Let’s discuss the Distributive Property in a bit more detail!**
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Distributive Property
The distributive property will come up time and time again, which highlights how
important it is. The property states: a(b + c) = ab + ac for integers a, b, and c
a) factored to expanded form
b) expanded to factored form
Let’s use area models to better understand this property. If a, b and c are positive integers
such that a and b + c represent the length and width of a rectangle, then the area of
the rectangle is given by:
A = l · w = a (b + c)
We can also look at this in a different way. The area of this rectangle is just the sum of
the areas of two smaller rectangles!
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Order of Operations
Order of Operations
Previously we’ve learned the following operations: addition, subtraction, multiplication (repeated addition),division, and exponentiation (repeated multiplication).
Now, suppose we were given the following exercise:
2 + 3(4 + 2 − 6 ÷ 2)2 − 3 · 8 ÷ 23
One very reasonable question to ask is: “Does the order in which I perform the
operations matter, and if so, where do I begin?” Well, the order is actually very
important! So to help us remember, a mnemonic device (BEDMAS) has been created. It
is described below:
Now we are equipped to create a step-by-step solution:
2 + 3(4 + 2 − 6 ÷ 2 )2 − 3 · 8 ÷ 23 = 2 + 3( 4 + 2 − 3)2 − 3 · 8 ÷ 23
= 2 + 3( 6 − 3 )2 − 3 · 8 ÷ 23
= 2 + 3 (3)2 − 3 · 8 ÷ 23
= 2 + 3(9) − 3 · 8 ÷ 23
= 2 + 3(9) − 3 · 8 ÷ 8
= 2 + 27 − 3 · 8 ÷ 8
= 2 + 27 − 24 ÷ 8
= 2 + 27 − 3
= 29 − 3
= 26
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Divisibility Rules
Let’s play a game using the rules below!
Number Divisibility Rule The number must be even (i.e. the last digit is 0, 2, 4, 6 or 8). The sum of the digits must be divisible by 3. 2 3 4 The last two digits must be divisible by 4 (notice: 4 = 2! ). The last digit must be 𝟎 or 𝟓. The number must be divisible by 𝟐 and 𝟑. When the last digit is removed, doubled and subtracted from the remaining number, the result must be divisible by 7. 5 6 7 8 9 The last three digits must be divisible by 8 (notice: 8 = 2! ). The sum of the digits must be divisible by 9. 10 11 The last digit must be 𝟎. After adding every other digit (starting with the second digit), and subtracting from it the sum of the remaining digits, the result must be divisible by 11. The number must be divisible by 𝟑 and 𝟒. 12 Example e.g. 2345678 e.g. 357492, since 3 + 5 + 7 + 4 + 9 + 2 = 30, which is divisible by 3! e.g. 249801275612 e.g. 24980127565 e.g. 2005464 Note: 2 + 5 + 4 + 6 + 4 = 21 e.g. 1092 Double last digit: 2(2) = 4 Remaining number: 109 Subtract: 109 − 4 = 105 Repeat! e.g. 249801275064 e.g. 100070001 since 1 + 7 + 1 = 9, which is divisible by 9! e.g. 1234567890 e.g. 150909 since (5 + 9 + 9) − (1 + 0 + 0) = 22, which is divisible by 11! e.g. 20001108 Note: 2 + 1 + 1 + 8 = 12, which is divisible by 3! [email protected]
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Factors
1. divides (verb):
If a and b are positive integers, we say that “b divides a” if we can find some positive
integer c, such that a = b · c (i.e. there is no remainder when a is divided by b). If c
exists, then it also divides a. We will refer to b and c as a “pair” that makes a.
For example, 7 divides 28 because we can find a positive integer (namely 4) that
we can multiply with 7 in order to get 28!
Here a =
,b=
, and c =
.
2. factor (noun):
If a and b are positive integers, we say that “b is a factor of a” if b divides a.
Therefore, by the definition of “divides,” there exists some positive integer c (that is
also a factor), such that a = b · c.
For example, 5 is a factor of 15 because we can find a positive integer (namely 3)
that we can multiply with 5 in order to get 15!
,b=
, and c =
.
Here a =
3. common factor:
If a, b and c are positive integers, we say that “c is a common factor of a and b” if
c is a factor of a and c is a factor of b.
For example, 3 is a common factor of 12 and 18 because we can find the couples
( 3 , 4 and 3 , 6) that make 12 and 18!
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Primes
1. prime:
A prime number is a positive integer whose only factors are 1 and itself.
e.g. 2, 3, 5, 7, 11, 13, 17, 19, 23, etc.
2. composite:
A composite number is a positive integer that is not prime (i.e. It has at least one
factor in addition to 1 and itself. e.g. 4, 6, 8, 9, 10, 12, 14, 15, 16, etc.
3. unit:
The number “1” is neither prime nor composite, and is called a unit.
1. Fundamental Theorem of Arithmetic:
Every positive integer greater than 1 is either prime, or can be expressed as a product
of primes in a unique way (except for a change in order). This expression of a number
as a product of primes is called the prime decomposition.
Use a factor tree and exponents to find the prime decomposition of 60.
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GCF
1. Greatest Common Factor (GCF):
If a and b are positive integers, the greatest common factor of a and b (denoted
by GCF(a, b)) is the largest common factor of a and b.
Find the greatest common factor of 36 and 84.
One strategy would be to list all of the factors of 36 and 84, and find the largest one that
they have in common: 36 : 1, 2, 3, 4, 6, 9, 12 , 18, 36
84 : 1, 2, 3, 4, 6, 7, 12 , 14, 15, 20, 30, 60
Here we can see that the GCF(36, 84) = 12. However, listing factors can become very
time consuming, especially in the case of larger numbers. Question: Is there a more
efficient method for finding the GCF? Answer: Yes!
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LCM
1. multiple:
If a and b are positive integers, we say that b is a multiple of a if a is a factor of
b (i.e. b = ac for some positive integer c). For example, multiples of 6 include
6, 12, 18, 24, 30, 36, 42, 48, etc.
2. Least Common Multiple (LCM):
If a and b are positive integers, the least common multiple of a and b (denoted
by LCM(a, b)) is the smallest multiple that has a and b as factors. For example,
below we can see that of the common multiples in bold of 6 and 9, 18 is the smallest:
6 : 6, 12, 18 , 24, 30, 36, 42, 48, 54, . . .
9 : 9, 18 , 27, 36, 45, 54, . . .
Find the least common multiple of 6 and 9 efficiently.
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Let’s Play!
1. Express in expanded form and calculate:
(a) 52 , 102 , 33 , and 24
(b) 5(2 + 8) and 11(3 + 4)
(c) Challenge! Expand and simplify: 3(k + 2) and t(4 + 7)
2. Calculate the following:
(a) 3 · 23 ÷ 6 + (3 · 4 − 32 ) + (42 − (8 + 2))
(b) 1 + (22 − 1)2 − 62 ÷ 9 + 2(5 − 22 )
3. Using divisibility rules and your knowledge of factors:
(a) Find all of the factors (between 1 and 12) for the following numbers:
i) 252
ii) 840
iii) 1386
iv) 22743
v) 620705910 vi) 1012006545
(b) Challenge! Find factor pairs from the factors found in part (a). You may want
to grab a calculator for the really large numbers (although you can also use long
division if you’re feeling ambitious)!
4. Use factor trees
a) 25
b) 48
f) 40
g) 64
k) 87
l) 91
to find the prime decompositions of the following:
c) 56
d) 63
e) 72
h) 55
i) 82
j) 75
m) 99
n) 108
5. Find the GCF and LCM of the following pairs. Feel free to use some of the results
in the question 4.
a) 12, 18
b) 7, 11
c) 48, 56
d) 6, 35
e) 64, 72 f) 55, 75
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