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Year 8 Mathematics
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Multiples, Factors and Primes
Learning Intentions
• Learning Intentions
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Understand what is meant by the terms Factor and Multiple
Be able to find the factors of a number
Be able to find the prime factors of a number
Be able to write a number as a product of its prime factors
Be able to find the common factors of two or more numbers
Be able to find the highest common factor (HCF) between two or more
numbers
– Be able to find the multiples of a number
– Be able to find the common multiples of two or more numbers
– Be able to find the lowest common multiple (LCM) between two or
more numbers
Factors of a Number
• A factor of a number is a number that divides
evenly into it.
• For example, the factors of 6 are 1, 2, 3 and 6.
• Each of the numbers 1, 2, 3 and 6 divide
evenly into 6.
• We can express 6 as a product of its factors
6 = 1 x 6
6 = 2 x 3
Prime Factors
• The prime factors of a number are the factors
of a number that are also prime numbers.
• For example:
– The factors of 6 are 1, 2, 3 and 6.
– The prime factors of 6 are 2 and 3.
Eratosthenes of Cyrene (275-194 B.C)
• Eratosthenes was a prominent Greek scholar who spent his early life in
Athens.
• He excelled in many areas, notably mathematics, astronomy, geography,
history, poetry and athletics.
• He was a universal genius who was known to his friends as Beta, because
he was regarded as the second best in almost all the fields he studied.
• He eventually went to Alexandria (Egypt) where he became the 3rd
librarian at the great university as well as private tutor to the son of
Ptolemy III.
• It was Eratosthenes who suggested a calendar
(later adopted by the Romans) of 365 days with an
additional day every 4th year.
• During old age he went blind and ended his life
• by drinking poison.
Eratosthenes
• Eratosthenes is best remembered today for
two notable achievements:
• The use of his “Sieve” to isolate prime
numbers
• His ingenious method for determining the
distance around the Earth with a high degree
of accuracy.
Strike out the twos, threes, fives and sevens from the Sieve of Eratosthenes
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Finding the Prime Factors
• We can find the prime
factors of a number using
the following method.
• We divide the number by
the smallest prime number
that is a factor
• We divide the answer again
by the smallest prime factor
• We repeat this process until
we end up at the number 1
2
120
2
60
30
Product of Prime Factors
• Once we have divided the number by the prime
factors we can write the number as a product of
the prime factors
• For example
6=2x3
 120 = 2 x 2 x 2 x 3 x 5
• Since 2 is multiplied by itself in the final sum, we
can simplify this to:
 120 = 23 x 3 x 5
• This is referred to as INDEX NOTATION.
Common Factors
• Sometimes numbers have factors in common
• For example, the numbers 12 and 24 both
have 2 as a common factor.
• However, they also have 3, 4, 8 and 12 as
common factors.
• 12 is the Highest Common Factor since it is
the largest factor that both numbers have.
Finding the HCF
• We can use the previous
technique for finding the HCF.
• For example, find the HCF of
16 and 36
• We begin by finding the prime
factors of both numbers
• Next, we find the common
prime factors
• Finally we multiply one set of
the numbers together
• The HCF is 2 x 2 = 4
2
16
2
36
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8
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18
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3
9
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1
Multiples
• A multiple of a number is a number that the
first number divides into.
• For example, some of the multiples of 6 are:
– 6, 12, 18, 24, 30, …
Common Multiples
• Common multiples are multiples that are
common to two or more numbers.
• For example, common multiples of 6 and 8
are:
– 24, 48, 72, 96, …
Finding Common Multiples
• To find the common multiples of two or more
numbers we list all the multiples and look for
the multiples that are in common.
• For example: find the common multiples of 4
and 6.
 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, …
 6: 6, 12, 18, 24, 30, 36, 42, 48, …
LCM
• The Lowest Common Multiple (LCM) is the
smallest multiple of two or more numbers.
• Fore example, find the LCM of 12 and 30
Venn Diagram
• Venn diagrams were invented by John Venn in 1880.
• They consist of a rectangular box with circles inside it.
• Each circle represent the values in one set and the
overlap represents values that appear in both sets.
• For example:
– One circle could represent the prime factors of the
number 12
– Another circle could represent the prime factors of the
number 30
– The overlap would represent the numbers that were prime
factor of both 12 and 30.
Using a Venn Diagram
• We can use the Venn diagram to help. We
place the common factors in the middle and
the others in the appropriate circles.
12
30
2
3
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5
Finding the LCM
• We find the LCM by multiplying all the
numbers in the diagram.
• LCM = 2 x 2 x 3 x 5 = 60