Transcript Multiples

Multiples
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The multiples of any given number are
0011 0010 1010
1101 0001
0100 1011 the number in turn
obtained
by multiplying
by each of the natural numbers 1, 2, 3, 4, 5, 6
and so on. For example
• Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24,
27, 30, 33, 36, 39, 42, 45, 48, …
• Multiples of 7 are 7, 14, 21, 28, 35, 42, 49,
56, 63, 70, 77, 84, 91, …
• Multiples of 37 are 37, 74, 111, 148, 185,
222, 259, 296, 333, 370, …
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Natural Numbers
0011 0010 1010 1101 0001 0100 1011
• It should be made clear that in this presentation we
will deal only with what are called the natural
numbers. These are the numbers we use for
counting: 1, 2, 3, 4, 5, 6, 7 and so on, ad
infinitum. So when you read the word ‘number’
in this presentation it refers to one of these. We
are therefore excluding zero, negative numbers
and anything other than positive whole numbers.
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0011 0010 1010 1101 0001 0100 1011
• You can easily generate the multiples of a given
number using a simple calculator with a constant
facility (most have this). Enter the number, for
example, 18, on to the calculator, press “+”, then
repeatedly press the equals (=) button. The
calculator responds by repeatedly adding 18 to
itself and thus produces the multiples of 18:
18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, …
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The Transitive Property of Multiples
0011 0010 1010 1101 0001 0100 1011
• The mathematical relationship “is a multiple
of” applied to numbers possesses what is
called the transitive property. Formally,
this means that if A is a multiple of B and B
is a multiple of C then it follows that A is a
multiple of C.
• 24 is a multiple of 6 and 6 is a multiple of 3,
therefore 24 must also be a multiple of 3.
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The Transitive Property of Multiples
0011 0010 1010 1101 0001 0100 1011
is a multiple of
A
is a multiple of
B
24
6
is a
multiple
of
is a
is a
multiple
multiple
of
is a
multiple
of
of
C
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2
3
4
Applying this principle, we can deduce that all multiples
of 6 are multiples of 3 (but not vice versa). Similarly,
all multiples of 28 must be multiples of 7, because 28 is
itself a multiple of 7.
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Multiples
• Being able to recognize multiples and having an
awareness of some of the patterns and
relationships within them help to develop a high
level of confidence and pleasure in working with
numbers. For example, this pattern in the
multiples of 37 may appeal to some students:
0011 0010 1010 1101 0001 0100 1011
3 x 37 = 111
6 x 37 = 222
9 x 37 = 333
12 x 37 = 444 and so on.
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Multiples
• The patterns in the multiples of 9, with the tens
digits increasing by one and the units digit
decreasing by one each time, is a useful aid for
learning the 9-times tables:
0011 0010 1010 1101 0001 0100 1011
1 x 9 = 09
2 x 9 = 18
3 x 9 = 27
4 x 9 = 36
5 x 9 = 45
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and so on.
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0011 0010 1010 1101 0001 0100 1011
View video clip at
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http://au.youtube.com/watch?v=oamy8L2lDZM&feature=related
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before proceeding further please.
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Multiples of 10
0011 0010 1010 1101 0001 0100 1011
• It is obvious at a glance that all these
numbers are multiples of ten:
20, 450, 980, 7620.
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• All multiples of ten end with the digit
zero.
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Multiples of 2
0011 0010 1010 1101 0001 0100 1011
• All multiples of two (even
numbers) end in 0, 2, 4, 6, or 8.
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Multiples of 5
0011 0010 1010 1101 0001 0100 1011
• All multiples of 5 end in either 0 or 5.
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2
• The multiples of 5 are 5, 10, 15, 20, 25,
30, 35, 40, 45, 50, 55, 60, 65, 70, 75,
80, 85, 90, 95, 100, 105, 110, 115, …
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Multiples of 4
0011 0010 1010 1101 0001 0100 1011
• There’s a simple way to spot whether a number
greater than 100 is a multiple of 4. Since 100 is a
multiple of 4, then any multiple of 100 is a
multiple of 4. So, given for example 4 528, we
can think of it as 4 500 + 28. We know that 4 500
is a multiple of 4, because it’s a multiple of 100.
So all we need to decide is whether the 28 is a
multiple of 4, which it is. So if you have a
number with three or more digits you need only
look at the last two digits to determine whether or
not you are dealing with a multiple of 4.
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Multiples of 9
• Would you spot immediately that all these are multiples of
0011 0010
1010
nine:
18,1101
72, 0001
315, 0100
567, 1011
4 986? If so, you may be making use
of the digital sum for each number. This is the number you get
if you add up the digits in the given number. If you then add
up the digits in the digital sum, and keep going with this
process of adding the digits until a single-digit answer is
obtained, the number you get is called the digital root. For
example, 4 986 has a digital sum 4 + 9 + 8 + 6 = 27. This is
itself a multiple of nine! This is true for any multiple of 9: the
digital sum is itself a multiple of 9. If you then add up the
digits of this digital sum (2 + 7) you get the single-digit
number, 9, which is therefore the digital root. Fascinatingly,
the digital root of a multiple of 9 is always 9.
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Spotting Multiples: Summary
• Every natural number is a multiple of 1.
0011 0010 1010 1101 0001 0100 1011
• Multiples of 2 are even numbers and end in 0, 2, 4, 6 or 8.
• The digital sum of a multiple of 3 is always a multiple of 3; and
the digital root is always 3, 6 or 9.
• The number given by the last two digits of a multiple of 4 must
be a multiple of 4.
• Multiples of 5 always end in 0 or 5.
• The number given by the last three digits of a multiple of 8 must
be a multiple of 8.
• The digital sum of a multiple of 9 is always a multiple of 9; and
the digital root is always 9.
• Multiples of 10 always end in zero.
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LCM – Lowest Common Multiple
• If we list all the multiples of each of two numbers, then
0011 0010 1010 1101 0001 0100 1011
inevitably there will be some multiples common to the two sets.
For example, with 6 and 10, we obtain the following sets of
multiples:
Multiples of 6:
Multiples of 10:
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2
6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, …
10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, …
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• The numbers common to the two sets are 30, 60, 90, 120, and
so on. The smallest of these (30) is known sometimes as the
lowest common multiple. The lowest common multiple of 6
and 10 is therefore the smallest number that can be split up into
groups of 6 and into groups of 10.
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Practical Applications of LCMs
• A class of 30 children is the smallest class-size that can be
organised into groups of 6 children for mathematics and teams of
10 for games.
• If I have to feed one plant every six days and another plant every
ten days, then the lowest common multiple indicates the first day
on which both plants have to be fed, namely the thirtieth day.
• If I can buy a certain kind of biscuit only in packets of 10 and I
want to share the biscuits equally between 6 people, the number
of biscuits I buy must be a multiple of both 6 and 10; so the
smallest number I can purchase is the lowest common multiple,
which is 30 biscuits.
0011 0010 1010 1101 0001 0100 1011
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1. Given 37 x 3 = 111, what is the
missing number in 37 x ? = 999.
0011 0010 1010 1101 0001 0100 1011
a)
b)
c)
d)
e)
9
12
18
27
33
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1. Given 37 x 3 = 111, what is the
missing number in 37 x ? = 999.
0011 0010 1010 1101 0001 0100 1011
a)
b)
c)
d)
e)
27
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2. Which of the following numbers
is a multiple of 9?
0011 0010 1010 1101 0001 0100 1011
a)
b)
c)
d)
e)
47
172
2 401
2 652
6 570
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2. Which of the following numbers
is a multiple of 9?
0011 0010 1010 1101 0001 0100 1011
a)
b)
c)
d)
e)
6 570
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3. What is the smallest number of people
that can be split up equally into groups of 8
and groups of 12?
0011 0010 1010 1101 0001 0100 1011
a)
b)
c)
d)
e)
12
16
24
48
96
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3. What is the smallest number of people
that can be split up equally into groups of 8
and groups of 12?
0011 0010 1010 1101 0001 0100 1011
a)
b)
c)
d)
e)
24
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4. Find a number that is the lowest
common multiple of all the numbers from 1
0011
1101 0001 0100 1011
to0010
10 1010
inclusive.
a)
b)
c)
d)
e)
55
2 520
7 560
8 640
3 628 800
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4. Find a number that is the lowest
common multiple of all the numbers from 1
0011
1101 0001 0100 1011
to0010
10 1010
inclusive.
a)
b)
c)
d)
e)
2 520
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5. Mrs. Hernandez waters one of her plants every 10 days
and another plant every 14 days. If she waters both plants
today, when is the next time both plants will be watered
on the same day?
0011 0010 1010 1101 0001 0100 1011
a)
b)
c)
d)
e)
in 14 days
in 28 days
in 35 days
in 70 days
in 140 days
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5. Mrs. Hernandez waters one of her plants every 10 days
and another plant every 14 days. If she waters both plants
today, when is the next time both plants will be watered
on the same day?
0011 0010 1010 1101 0001 0100 1011
a)
b)
c)
d)
e)
in 70 days
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References
0011 0010 1010 1101 0001 0100 1011
• Derek Haylock, Mathematics Explained for
Primary Teachers (2nd edition, 2002),
Chapter 11, Pages 111 – 114, 118 – 119.
• http://au.youtube.com/watch?v=oamy8L2lD
ZM&feature=related
• http://www.mathgoodies.com/lessons/vol3/l
cm.html
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