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Shania QQ:1246640685MSN: [email protected]
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Contents
Math words quiz
Factors-prime factors
Multiples-LCM
Patterns and sequences
SETS!!!
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Factors
Factors of a number are the whole numbers that multiply together to give the original number
E.g. The factors of 12 are? 12 is the the original number So which numbers can multiply
together to give 12? 1×12, 2×6, 3×4 That is, 1,2,3,4,6,12 are factors of
12. We use F(12) as a short way of
writing factors of 12. F(12)={1,2,3,4,6,12} Factor pairs of 12 are (1,12), (2,6),
(3,4)
Among these 6 factors 2 and 3 are prime factors.
[Prime factors of a number are factors of the number that are also prime number.]
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Writing numbers as the product of prime factors
Prime factors2,3,5,7,11,13…12=4×3,but 4 is not prime number,
we break 4 down further 12=2×2×3 that we have written 12 as the product of prime factors
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•Step 1
• Step 2
•Step 3
Try to divide the given number by the first prime number -- 2
Continue until 2 will no longer divide into it
Try the next prime number, 3, then 5, 7 and so on, until final answer is 1
Steps
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Examples
Write 60 as a product of prime factors.
Write 3465 as a product of prime factors.
So 3465=3×3×5×7×11
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Several rules
The number is ended by 0,2,4,6,8 can be divided
by 2.
The sum of all digits of the number can be divided
by 3, that is, the number can be divided by 3.
3465 3+4+6+5=18 18/3=6
so 3465 can be divided by 3
So does 5, 7, 11 and 13
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Multiples
Definition: the multiples of number are the products of that numbers and 1,2,3,4,5…(Natural number)
E.g. The multiples of 3 are??? 3, 6, 9, 12, 15… The first five multiples of 3:
M(3)={3,6,9,12,15}
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LCM-Lowest common multiple
最小公倍数The smallest number that is a
multiple of two or more numbers12, 24, 36 are multiples of 3 and 4.
BUT, 12 is the smallest one, that is, 12 is the LCM of 3 and 4.
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Two ways to find LCM
ONE: ①List the multiples of each numbers of each numbers ②and then pick out the lowest number that appears in every one of the lists. (applicable for small numbers)
ANOTHER: Expressing each number as a product of prime factors
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Sets
Any collection of objects – have sth in common, some connection with each other.
{ } braces , comma
The object in a set we called element of the set ∈
5 ways to express sets
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5 ways
1. Listed set {1,2,3,4.5}2. Described set {first five natural
numbers}3. Set builder notation to describe
sets mathematically {x:x≦10 and x is an even number}
4. Represented by a name or a letter {red, blue, yellow} {Thomas, Joise}
5. Venn diagram
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Special sets
Finite sets and infinite sets
Universal set rectangular Venn diagram
It can change from problem to problem
{ } empty set
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Relationships between sets
Equal sets: same cardinality and same elements “=“
Equivalent sets: same number of elements
Subsets: A is the subset of set B if all of the elements of A are elements of B
A⊂ B (子集) B⊃ A (superset 扩散集) In our book , different from Chinese book
How many subsets? Include {} and equal set Use permutation and combination to prove.
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Continue
Complement set A’ contains all of the elements of the universal set not in A. set A and its complement A’ are disjoint- A∩A’=empty set
Power set: All subsets of a given set A
If a set has n elements it will have 2^n subsets
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Intersection and union of sets
A∪B : the union of sets A and B.
A∩B : the intersection of sets A and B. The elements common to set A and B.
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Laws
1. A ∩ A = A2. A ∩ B = B ∩ A (commutative law) 3. A ∩ B ∩ C = A ∩ (B ∩ C) (associative la
w) 4. A ∩ φ = φ ∩ A = φ 5. A ∪ (A ∩ B) = A 6. A ∩ (A ∪ B) = A 7. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (distri
butive law) 8. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (distri
butive law)