1.1.5 Prime Factor Decomposition
Prime factors:
A prime factor is a factor which is also a prime number.
All natural numbers except \( 1 \) can be expressed as a product of prime factors.
e.g.
\( 21=3×7 \), \( 64=2\times 2\times 2\times 2\times 2\times 2 \), \( 24=2\times 2\times 2\times 3 \)
To find prime factors of a natural number we use factor tree method
- Split the original number into two branches by writing a pair of factors at the end of the branch
- If there is a prime number at the end of each branch, we have constructed a prime factor tree
- If not, repeat the process until we get prime number at the end of each branch
Worked Example:
Find the prime factors of \( 56 \).
Split \( 56 \) in two parts and write at the end of branches
\( 2 \) is a prime number, make a circle around it
Repeat the process for \( 28 \)
Continue for \( 14 \)
As, all the end numbers are circled, so it’s completed.
List every number that has been circled.
\( 56=2\times 2\times 2\times 7 \)
Write any repeated numbers as powers of the original number.
\( 56=2^3 \times 7 \)
Prime factorization can be used to find if a given number is a square number or a cube number
- A number is a square number if all the indices in the prime factorization of the number are even.
- A number is a cube number if every index in its prime factor decomposition is a multiple of three.
Worked Example:
\( N=2^4 \times 3\times7^5 \)
\( PN = K \), where \( P \) is an integer and \( K \) is a square number.
Find the smallest value of \( P \).
Substitute \( N=2^4 \times3 \times7^5 \) in the formula \( PN=K \)
\( P (2^4 \times 3 \times 7^5 )=K \)
\( 2,3,7 \) are all prime numbers, so for \( P (2^4 \times 3 \times 7^5 ) \) to be a square number,all the indices should be even
So smallest possible integer value of P is \( (3 \times 7 ) \)
\( P=21 \)
\( P=21 \)
Highest common factor (HCF):
The highest common factor of all the given numbers is the largest number which is a factor of all those numbers.
To find the highest common factor of given numbers
- Find prime factors of the given numbers
- List all the prime factors of the given numbers
- Multiply the common factors together
Worked Example:
Find the highest common factor of \( 84 \) and \( 105 \)
Find prime factors of \( 84 \) and \( 105 \)
List all the factors
\( 84=2\times 2\times 3\times 7 \)
\( 105=5\times 3\times 7 \)
Common multiple:
A number that occurs in both of two numbers’ times tables is referred to as a common multiple of those two numbers.
e.g.
The multiples of \( 3 \) are \( 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 \)
The multiples of \( 5 \) are \( 5, 10, 15, 20, 25, 30 \)
So common multiples of \( 3 \) and \( 5 \) are \( 15, 30 \)
Least common multiple (LCM):
A least common multiple of all the given numbers is the smallest number which is a multiple of all those numbers.
To find the least common multiple of the given numbers
- Find prime factors of the given numbers
- List all the prime factors
- Multiply the common factors and non-common factors together
Worked Example:
Find the least common multiple (LCM) of \( 56 \) and \( 42 \)
List all the factors
\( 56=2\times 2\times 2\times 7 \)
\( 42=2\times 3\times 7 \)
Multiply the common factors
\( 2\times 7=14 \)
Multiply the non-common factors
\( 2\times 2\times 3=12 \)
Now multiply the common factors and non-common factors together
\( 14\times 12=168 \)
Thus, the (LCM) of \( 56 \) and \( 42 \) is \( 168 \)
Least Common Multiple – Exercise
Lowest Common Multiple (LCM) Practice
Test yourself
Question 1:
Find the highest odd number which is a factor of \( 60 \) and \( 90 \).
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Question 2:
Find the lowest common multiple (LCM) of \( 20 \) and \( 24 \).
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Question 3:
Write down all the factors of \( 1028 \)
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Question 4:
\( 234=2 \times3^2 \times13 \), \( 1872=2^4 \times3^2 \times13 \)
\( 234 \times1872=438084 \)
Use this information to write \( 438084 \) as a product of its prime factors
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