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Factors
What is a factor?
concept of factor is simply the reverse
of multiple. If A is a multiple of B, then B
is a factor of A.
► For example, 24 is a multiple of 6, so 6 is a
factor of 24. Colloquially, we say “6 goes
into 24”.
► The factors of 24 are all those natural
numbers by which 24 can be divided
exactly: 1, 2, 3, 4, 6, 8, 12 and 24.
► The
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Recognizing quickly all the factors of a
given number is, of course, very useful.
►A
set of 24 cans can be put into groups of 2, 3, 4,
6, 8 or 12, as shown in the rectangular arrays
below:
OOOOOOOOOOOO
OOOOOOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOOOO
OOOOOO
OOOOOO
OOOOOO
OOOOOO
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The Transitive Property
►Similarly
to “is a multiple of”, the
mathematical relationship “is a
factor of” also possesses the
transitive property. For example,
any factor of 12 must be a factor of
24, because 12 is a factor of 24.
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Highest Common Factor (HCF)
► This
is a similar idea to the lowest common
multiple. If we list all the factors of two
numbers, the two sets of numbers may
have some numbers in common. (Since 1 is
a factor of all numbers, they must at least
have this number in common!) The largest
of these common factors is called the
highest common factor (HCF) or greatest
common factor (GCF).
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Highest Common Factor (HCF)
► For
example, with 24 and 32 we have the
following two sets of factors:
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
Factors of 32: 1, 2, 4, 8, 16, 32.
► The
factors in common are 1, 2, 4 and 8.
So the highest common factor is 8.
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What is a Prime Number?
► The
strict mathematical definition is that
any number that has precisely two factors,
and no more than two, is called a prime
number.
► In practice, we think of a prime number as
a number that cannot be divided exactly by
any number apart from 1 and itself.
► The prime numbers are 2, 3, 5, 7, 11, 13, …
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The Sieve of Eratosthenes
View the video clip here from
http://au.youtube.com/watch?v=9m2cd
WorIq8
please.
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Product of Prime Factors.
►24
can be written as a product of
factors in a number of different ways:
1 x 24, 2 x 12, 3 x 8, or 4 x 6.
►As a product of Prime Factors, there is
only one combination that will produce
24:
2 x 2 x 2 x 3.
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Using Prime Factors to find a HCF.
What is the HCF of 72 and 96?
►72
=2x2x2x3x3
►96 = 2 x 2 x 2 x 2 x 2 x 3
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Using Prime Factors to find a HCF.
►72
=2x2x2x3x3
►96 = 2 x 2 x 2 x 2 x 2 x 3
►72 = 24 x 3; 96 = 24 x 4.
►The HCF of 72 & 96 = 24
►24 = 2 x 2 x 2 x 3
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Another shortcut to HCF
► What
is the HCF of 75 and 125?
► Any factor of 75 and 125 must also be a
factor of (125 – 75), i.e. 50.
► Let’s start with 50, test if it is the required
number, then progress down to the next
highest factor of 50 (25, 10, 5, 2, 1) until
we arrive at the required answer.
► 50 won’t “go into” either 75 or 125, but 25
will, therefore 25 is our required answer.
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What is the HCF of 48 and 80?
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What is the HCF of 48 and 80?
48 = 2 x 2 x 2 x 2 x 3
80 = 2 x 2 x 2 x 2 x 5
HCF of 48 and 80 = 2 x 2 x 2 x 2 = 16
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The year 2003 was a prime year. When
next will the year be a prime number?
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The year 2003 was a prime year. When
next will the year be a prime number?
Not 2004 (even number)
Not 2005 (divisible by 5)
Not 2006 (even number)
Not 2007 (2 + 0 + 0 + 7 = 9, therefore 2007 is
divisible by both 3 and 9)
Not 2008 (even number)
Not 2009 (= 7 x 287)
Not 2010 (divisible by 2, 5 and 10)
2011 looks a likely answer!
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Which of the following is
equivalent to 144?
a)
b)
c)
d)
e)
22
3
2
3
2
24
24
x
x
x
x
x
34
2
3
3
3
32
33
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Which of the following is
equivalent to 144?
a)
b)
c)
d)
e)
2 4 x 32
-
( 16 x 9 = 144 )
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Which of the following numbers
is not a prime number?
5
17
31
47
7
19
37
49
11
23
41
53
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13
29
43
59
19
Which of the following numbers
is not a prime number?
5
17
31
47
7
19
37
49
11
23
41
53
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13
29
43
59
20
What is the most number of rows in which 120 chairs
(45 red and 75 blue) can be arranged so that each row
has the same numbers of each colour?
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What is the most number of rows in which 120 chairs
(45 red and 75 blue) can be arranged so that each row
has the same numbers of each colour?
uuu
uuuuu
uuu
uuuuu
uuu
uuuuu
uuu
uuuuu
uuu
uuuuu
uuu
uuuuu
uuu
uuuuu
uuu uuuuu
…
uuu
uuuuu
15 rows, each of 3
red and 5 blue, gives
45 + 75 = 120 chairs
in total.
Note that 15 is the
highest common
factor of 45 and 75.
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References
► Mathematics
Explained for primary teachers
(third edition, 2006) by Derek Haylock.
► http://au.youtube.com/watch?v=9m2cdWor
Iq8
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