Download unit 2 vocabulary: divisibility

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UNIT 2 VOCABULARY: DIVISIBILITY
1.1. Multiples and factors
We say that a number b is a factor or a divisor of another number a if the division a : b is an
exact division.
If a number can be expressed as a product of two natural numbers, then these numbers are called
factors of that number.
For example, 2 is a divisor of 6 and 3 is another divisor of 6.
2 and 3 are factors of 6.
Let's see a real life example of this.
A teacher wants to form groups with the same number of people in a class with 18 students. How can
he do it?
Let's start dividing in order:
18 ÷ 1 = 18, so he can make 1 group
with 18 students. 1 and 18 are
both factors of 18.
18 ÷ 2 = 9, so he can make 2
groups with 9 students. 2 and
9 are both factors of 18.
18 ÷ 3 = 6, so he can make 3
groups with 6 students. 3 and
6 are both factors of 18.
18 ÷ 4 is not an exact division. He
18 ÷ 5 is not an exact division.
can make 4 groups, but they don't
He can make 6 groups of 3, but
have the same number of students in He can make 5 groups, but
I already have 6 and 3 on my
they don't have the same
each group. 4 is NOT a factor of
number of students. 4 is NOT list of factors.
18.
a factor of 18.
As soon as you get factors that are already on your list, you can stop. All the factors you find are
repeats of factors you already have.
To sum up, let's organize this information:
Number of group (factors)
1
Number of students in each group 18
2
3
6
9
18
9
6
3
2
1
Can you see any curious pattern?
The products of a number with the natural numbers (1, 2, 3, 4, 5, ...) are called the multiples of the
number. The multiples of a number are obtained by multiplying the number by each of the natural
numbers.
For example, 0,3, 6, 9, 12, 15, 18 ... are multiples of 3.
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If a number "a" is a multiple of another number "b", then "b" is necessarily a factor
of "a".
For example, 20 is a multiple of 5, so 5 is a factor of 20.
Once again, let's see a real life example.
If you want to buy eggs, they are normally sold by dozens, and a dozen is 12. How many eggs can I
buy?
So the multiples of 12 are 12, 24, 36, 48, 60, 72, and so on.
Exercises.
1. Fill in the gaps with the words "multiple" or "factor":
7 is a __________ of 49
1 is a __________ of 17
49 is a __________ of 7
17 is a __________ of 1
2. Find three multiples of 11 that are between 27 and 90.
3. Work out if 556 is a multiple of 4.
4. Find out if 12 is a factor of 144.
5. Work out all the factors of the following numbers:
24
27
48
25
25 is a __________ of 100
13 is a __________ of 13
7
56
2.1. Prime and composite numbers
A prime number is a number that only has two factors: the number one and itself.
For example, 11 and 23 are prime numbers:
•
•
The only numbers that multiply to give 11 are 1 x 11.
The only numbers that multiply to give 23 are 1 x 23.
A composite number has more than two factors.
For example:
• 9 has three factors: 1, 3 and 9, so 9 is a composite number.
• 10 has four factors: 1, 2, 5 and 10, so 10 is a composite number.
1 is NOT a Prime Number – it Just Isn't!!!
1 is NOT a prime number – it doesn't have exactly to factors, and prime numbers do.
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A smart procedure to find the first prime numbers is the Sieve of Erathostenes*.
*Eratosthenes of Cyrene was a Greek mathematician, geographer, poet, athlete,
astronomer, and music theorist. He was the first person to use the word "geography"
in Greek and invented a system of latitude and longitude.
SIEVE OF ERATHOSTENES
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
You have to cross out using a red pen and circle using a green pen.
1. Cross out 1, as 1 isn't a prime number by definition.
2. Circle 2, as 2 is the smallest prime number.
3. Cross out all the multiples of 2 after 2, this is, all the even numbers.
4. Circle 3.
5. Cross out all the multiples of 3.
6. Circle 5.
7. Cross out all the multiples of 5, this is, all numbers ending in 0 or 5.
8. Circle 7.
9. Cross out all the multiples of 7.
10. You have now all the primes numbers up to 100!!!!
2.2. Divisibility tests
Divisibility rules help you check if one number is divisible by another, without having to do too much
calculation!.
Exercise. Watch this video and write a short summary using this table.
http://www.youtube.com/watch?v=UZYpOKtylxM
DIVISIBLE BY ...
2
3
4
IF
EXAMPLES
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5
6
9
10
Exercises.
1. Use the divisibility rules to complete (with true or false) the following table:
Divisible
by
2
3
4
5
9
10
25
100
375
990
1,848
12,300
14,240
2. Sort these numbers into 3 lists: multiples of 3, multiples of 4 and multiples of 5.
33
25
2.
3.
4.
5.
6.
1016
164
21
63
10
39
175
50
4036
51
35
11144
a) In the multiples of 5, what do you notice about the last digit?
b) In the multiples of 3, what do you notice about the digit sum?
c) In the multiples of 4, what do you notice about the last digit?
Is 3 a factor of 2001?
Find out which of the following numbers are prime:
71
77
83
107
117
Is 729,064 a prime number?
Express each of the following numbers as the sum of two prime numbers:
10
20
Which of these numbers are not prime? Show a factor for evidence.
221
35
784
20
97
110
512
30
2.3. Prime factor decomposition of a number
The "prime factorization or decomposition of a number" is finding which prime numbers multiply
together to make the original number.
For example, what is the prime decomposition of 12?
It is best to start working from the smallest prime number, which is 2, so let's
check:
12 ÷ 2 = 6
The remainder is 0, so we have taken the first step!
But 6 is not a prime number, so we need to continue. Let's try 2 again:
6÷2=3
Yes, the remainder is 0 again. And 3 is a prime number, so we have the answer: 12 = 2 · 2 · 3
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As you can see, every factor is a prime number, and 2 · 2 = 4, and 4 · 3 = 12, so the answer must be
right.
The factorization can also be written using exponents: 12 = 2² · 3
The prime factorization of a prime number is the same number!
For example, 17 is a prime number, so that is as far as we can go.
17 = 17
Exercise. Express the following as a product of prime factors:
7
9
47
105
648
220
405
864
1000
25920
3.1. Highest Common Factor
Factors that are common to two or more numbers are called common factors.
For example, let's consider numbers 8 and 12:
• The factors of 8 are 1,2 ,4 and 8.
• The factors of 12 are 1, 2, 3, 4, 6 and 12.
The common factors are those that are found in both lists:
• 1, 2 and 4 appear in both lists, so the common factors of 8 and 12 are 1, 2 and 4.
The greatest common factor of two or more numbers is called the Highest Common Factor (HCF).
For example, for numbers 8 and 12, the HCF is 4.
The "Highest Common Factor" is often abbreviated to "HCF", and is also known as:
• the "Greatest Common Divisor (GCD)", or
• the "Greatest Common Factor (GCF)"
Two numbers are relatively prime or coprimes if they have no common factors other than 1 (in other
words, their HCF is 1).
For example, 15 and 28 are relatively prime, because the factors of 15 (1,3,5,15), and the factors of
28 (1,2,4,7,14,28) are not in common (except for 1).
4.1. Lowest Common Multiple
Multiples that are common to two or more numbers are called common multiples.
For example, let's consider numbers 2 and 3:
• The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, ...
• The multiples of 3 are 3, 6, 9, 12, 15, 18, ...
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The common factors are those that are found in both lists:
• 6, 12, 18 ... appear in both lists, so the common multiples of 8 and 12 are 6, 12, 18...
• THEY ARE INFINITE!!!!!
The smallest common multiple of two or more numbers is called the lowest common multiple (LCM).
For example, for numbers 2 and 3, the LCM is 6.
The "Lowest Common Multiple" is often abbreviated to "LCM", and is also known as:
• the "Least Common Multiple (LCM)"
When two numbers are relatively prime, the least common multiple is the product of the two
numbers.
For example, 15 and 28 are relatively prime, so their LCM is 12 · 28 = 336
Exercises.
1. What is the LCM of 8 and 12?
2. What is the HCF of 14 and 21?
3. Find de HCF and the LCM of the following sets of numbers:
32 and 48
45 and 105
36, 84 and 132
4. Find the HCF and the LCM of 12, 18, 24 and 30.
5. There are two sets of traffic lights outside Eric's house. One day, he times how
often they change. Lights A turn green exactly every 60 seconds. Lights B turn
green exactly 70 seconds. At 6:00 pm precisely they both turn green together. At
what time do they both turn green together again?
6. In a school classroom there are 48 girls and 64 boys. The teacher
wants to put the students in equal rows. Only girls or boys can be in
each row. What is the greatest number of students that can be in each
row?