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Prime Factorisation
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Foundational skills
Recognise a valid prime factorisation
\[ 2^2 \times 3 \times 5 \text{ valid?} \]
Identify whether an expression is a valid prime factorisation.
Write repeated factors using index notation
\[ 2 \times 2 \times 2 \times 3 \times 3 \]
Convert repeated multiplication of primes into index form.
Expand index notation to repeated factors
\[ 2^3 \times 5^2 \rightarrow \text{?} \]
Convert index form back to repeated multiplication.
Calculate the value from index notation
\[ 2^3 \times 3^2 = \text{?} \]
Evaluate a number given in prime factorisation form.
Finding prime factorisations
Prime factorise a number up to 50
\[ 48 = \text{?} \]
Find the prime factorisation of a number up to 50.
Prime factorise a number 51 to 100
\[ 84 = \text{?} \]
Find the prime factorisation of a number between 51 and 100.
Prime factorise a number 101 to 200
\[ 180 = \text{?} \]
Find the prime factorisation of a number between 101 and 200.
Prime factorise a number over 200
\[ 360 = \text{?} \]
Find the prime factorisation of a larger number.
Prime factorise a product
\[ 12 \times 15 = \text{?} \]
Find the prime factorisation without first multiplying.
Complete a partially filled factor tree
\[ \text{Factor tree: } 72 \rightarrow ? \times 8 \]
Fill in missing values in a factor tree diagram.
Reading prime factorisations
Identify if a number is a factor
\[ 180 = 2^2 \times 3^2 \times 5, \text{ is 12 a factor?} \]
Determine if one number is a factor of another using prime factorisation.
List all factors from prime factorisation
\[ 12 = 2^2 \times 3 \rightarrow \text{list factors} \]
List all factors of a number given in prime factorised form.
HCF using prime factorisation
Find HCF when both numbers are factorised
\[ 36 = 2^2 \times 3^2, 60 = 2^2 \times 3 \times 5 \]
Find the HCF of two numbers already in prime factorised form.
Find HCF by prime factorising both numbers
\[ \text{HCF of } 48 \text{ and } 72 \]
Find the HCF by first finding their prime factorisations.
Find HCF of three numbers
\[ \text{HCF of } 24, 36, 60 \]
Find the HCF of three numbers using prime factorisation.
LCM using prime factorisation
Find LCM when both numbers are factorised
\[ 18 = 2 \times 3^2, 24 = 2^3 \times 3 \]
Find the LCM of two numbers already in prime factorised form.
Find LCM by prime factorising both numbers
\[ \text{LCM of } 28 \text{ and } 42 \]
Find the LCM by first finding their prime factorisations.
Find LCM of three numbers
\[ \text{LCM of } 12, 15, 20 \]
Find the LCM of three numbers using prime factorisation.
Applications
Simplify a fraction using prime factorisation
\[ \frac{84}{126} = \text{?} \]
Use prime factorisation to find the HCF and simplify fully.
Identify perfect squares
\[ 324 = 2^2 \times 3^4 \text{ square?} \]
Recognise that perfect squares have all even powers.
Identify perfect cubes
\[ 216 = 2^3 \times 3^3 \text{ cube?} \]
Recognise that perfect cubes have all powers divisible by 3.
Find smallest multiplier for a perfect square
\[ 50 = 2 \times 5^2 \rightarrow \times ? \]
Find what’s needed to make all prime factor powers even.
Find smallest multiplier for a perfect cube
\[ 72 = 2^3 \times 3^2 \rightarrow \times ? \]
Find what’s needed to make all powers divisible by 3.
Calculate the number of factors
\[ 60 = 2^2 \times 3 \times 5 \rightarrow ? \text{ factors} \]
Use the formula: add 1 to each power, then multiply.
Find number of trailing zeros
\[ 2^4 \times 5^3 \times 7 \rightarrow ? \text{ zeros} \]
Count trailing zeros by counting pairs of 2 and 5.
Problem solving
Find number from constraints
\[ \text{Multiple of 4, factor of 48, 6 factors} \]
Use prime factorisation to solve with multiple constraints.
Use HCF × LCM = a × b
\[ \text{HCF}=8, \text{LCM}=120, a=24 \]
Use the relationship HCF × LCM = product of two numbers.
Special cases
HCF of algebraic expressions
\[ \text{HCF of } 12a^2b \text{ and } 18ab^3 \]
Find the highest common factor of two algebraic expressions.
LCM of algebraic expressions
\[ \text{LCM of } 6x^2y \text{ and } 9xy^3 \]
Find the lowest common multiple of two algebraic expressions.
Find unknown powers in a prime factorisation
\[ 2^a \times 3^b = 72 \]
Find unknown powers given the primes and the value.
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