Download Factor trees, prime factors, highest common factors

17/08/2016
Prime factors
Lowest common multiples and
highest common factors
All whole numbers can be made by multiplying a combination of prime
numbers together
These prime factors can be found with a factor tree
2 is a prime number and is a factor of 48
If 48 is divided by 2, the answer is 24
48
2 is a factor of 24
2
24
If 24 is divided by 2, the answer is 12
2 is a factor of 12
If 12 is divided by 2, the answer is 6
12
2
6
2
2
3
2 is a factor of 6
If 6 is divided by 2, the answer is 3
3 is a prime number it cannot be divided
2 and 3 are the prime factors of 48
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There can be many different ways of setting out a factor tree, but they all
have the same outcome
Two different factor trees for 48
48
Neither of the factors on
the first branches of this
tree are prime
2
48
4
12
2
8
2
24
6
2
3
6
2
2
2
2
3
There is at least one prime
factor in every branch of this
tree
The result is the same
48 = 2 x 2 x 2 x 2 x 3
2 and 3 are the prime factors of 48
Two different factor trees for 50
50
2
50
25
5
5
5
10
5
2
50 = 2 x 5 x 5 = 2 x 52
2 and 5 are the prime factors of 50
The non-prime factors are:
2 x 5 = 10
5 x 5 = 25
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Using factor trees to find common factors and multiples
A factor tree for 72
A factor tree for 64
64
72
2
2
36
3
32
2
16
2
12
2
6
3
8
4
2
2
2
2
64 = 2 x 2 x 2 x 2 x 2 x 2 = 26
72 = 2 x 2 x 2 x 3 x 3 = 23 x 32
2 and 3 are the prime factors of 72
2 is the only prime factor of 64
Using factor trees to find common factors and multiples
2 is the only prime factor of 64
A factor tree for 64
64
6 2s form the branches of the tree,
so the non prime factors are:
2x2=4
2
32
16
2
2
8
2x2x2=8
2 x 2 x 2 x 2 = 16
2 x 2 x 2 x 2 x 2 = 32
2 x 2 x 2 x 2 x 2 x 2 = 64
4
2
2
2
64 = 2 x 2 x 2 x 2 x 2 x 2 = 26
click
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17/08/2016
Using factor trees to find common factors and multiples
A factor tree for 72
2 and 3 are the prime factors of 72
72
2
36
3
12
2
6
2
3
72 = 2 x 2 x 2 x 3 x 3 = 23 x 32
A factor tree for 360
The non-prime factors are:
2 x 2 = 4..
2 x 2 x 2 = 8..
2 x 3 = 6..
2 x 2 x 3 = 12
2 x 2 x 2 x 3 = 24
3 x 3 = 9..
3 x 3 x 2 = 18
3 x 3 x 2 x 2 = 36
3 x 3 x 2 x 2 x 2 x 2 = 72
A factor tree for 320
360
5
320
72
2
5
36
3
64
2
12
2
2
6
3
32
16
2
2
360 = 2 x 2 x 2 x 3 x 3 x 5 = 23 x 32 x 5
8
4
2
2
2
320 = 2 x 2 x 2 x 2 x 2 x 2 x 5 = 26 x 5
2, 3 and 5 are the prime factors of 360
2 and 5 are the prime factors of 320
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The Highest Common Factor
72 = 2 x 2 x 2 x 3 x 3 = 23 x 32
64 = 2 x 2 x 2 x 2 x 2 x 2 = 26
The only factors shared by 64 and 72 are 2 x 2 x 2
So the highest common factor of 64 and 72 is 2 x 2 x 2 = 8
The Highest Common Factor of 320 and 360
360 = 5 x 2 x 2 x 2 x 3 x 3 = 5 x 23 x 32
320 = 5 x 2 x 2 x 2 x 2 x 2 x 2 = 5 x 26
The only factors shared by 360 and 320 are 2 x 2 x 2 and 5
So the highest common factor of 360 and 320 is 5 x 2 x 2 x 2 = 40
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The LowestCommon Multiple (or Least Common Multiple)
72 = 23 x 32
64 = 26
the lowest common multiple is found by multiplying the highest power
of all of the factors of both numbers
32 Is the highest power of prime factor 3
26 Is the highest power of prime factor 2
So the lowest common multiple of 64 and 72 is 26 x 32 = 576
9 x 64 = 576
8 x 72 = 576
The Lowest Common Multiple of 320 and 360
360 = 5 x 23 x 32
320 = 5 x 26
the lowest common multiple is found by multiplying the highest power
of the factors in each equation
32 Is the highest power of 3
26 Is the highest power of 2
5 Is the highest power of 5
So the lowest common multiple of 360 and 320 is 26 x 32 x 5 = 2880
8 x 360 = 2880
9 x 320 = 2880
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Find the prime factors using factor trees
Examples of lowest common multiples
for 24 you get prime factors 2, 2, 2, 3
for 36 you will get 2, 2, 3, 3
the two 2s of 36 are less than the three 2s of 24, so take the three 2s (you
could think of the two 2s as hiding behind the three}
the one 3 of 24 is less than the two 3s of 36, so take the two 3s
multiply them together – 3 x 3 x 2 x 2 x 2 = 9 x 8 = 72, which is the LCM 3 x 24
= 72, 2 x 36 = 72
you have to use the other factors too if there are any
for 66 you will get prime factors 2, 3, 11
for 60 you will get 2 ,2, 5, 3
(two 2s trump the single 2 of 66, so use 2 x 2)
take the different factors 2, 2, 3, 5 and 11 and multiply them together = 660
which is the lowest (sometimes least) common multiple. 3 is a factors of
both numbers, so you only need to use it once.
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