Prime factorisation
$24$ boxes in a warehouse can be arranged in many ways: $2 \cdot 12$, $3 \cdot 8$, $4 \cdot 6\ldots$ But there is only one way to break it down into prime factors: it is its mathematical "fingerprint".
The fundamental theorem: a unique factorisation
State which number corresponds to each of the following prime factorisations: a) $2^{2} \cdot 3 \cdot 5$ b) $2^{3} \cdot 7$ c) $3^{2} \cdot 11$
State which number corresponds to each of the following prime factorisations:
- a) $2^{2} \cdot 3 \cdot 5$
- b) $2^{3} \cdot 7$
- c) $3^{2} \cdot 11$
Classify the following natural numbers into prime or composite. Remember that prime numbers are those that have only two factors: $1$ and themselves.…
Classify the following natural numbers into prime or composite. Remember that prime numbers are those that have only two factors: $1$ and themselves.
$14, 17, 21, 23, 27, 29, 33, 37$
The factor tree
A visual way to break a number down: you keep splitting the number into any two factors until every branch ends in a prime number.
Complete the following factor tree for the number $24$. You must find two numbers that multiply to give the level above, until all the numbers at the ends are prime.…
Complete the following factor tree for the number $24$. You must find two numbers that multiply to give the level above, until all the numbers at the ends are prime.
Finally, write down the factorisation of $24$ as a product of its prime factors.
A video game designer wants to organise a virtual item warehouse. To do so, they need to know how many factors the number of available boxes has, which is $360$. a) Find…
A video game designer wants to organise a virtual item warehouse. To do so, they need to know how many factors the number of available boxes has, which is $360$.
- a) Find the prime factorisation of $360$.
- b) Using the rule of exponents (add one to each exponent in the factorisation and multiply them), calculate the total number of factors that $360$ has.
The vertical bar technique
The quickest method for large numbers: you repeatedly divide by the smallest possible prime until you reach $1$.
Fill in the blanks in the following prime factorisation of a number using the vertical line method: Write down the final result of the factorisation using powers.
Fill in the blanks in the following prime factorisation of a number using the vertical line method:
Write down the final result of the factorisation using powers.
Find the prime factorisation of the following numbers. Express the final result as a product of powers: a) $36$ b) $45$ c) $80$
Find the prime factorisation of the following numbers. Express the final result as a product of powers:
- a) $36$
- b) $45$
- c) $80$
Using the factorisation to reason
Once you have factorised a number, you can answer questions about multiples and divisors without calculating its value.
Look at the prime factorisation of the following numbers: $A = 2^{2} \cdot 3 \cdot 7$ $B = 2 \cdot 3^{2} \cdot 5$ Without carrying out the multiplications, answer with…
Look at the prime factorisation of the following numbers:
$A = 2^{2} \cdot 3 \cdot 7$
$B = 2 \cdot 3^{2} \cdot 5$
Without carrying out the multiplications, answer with reasoning:
- a) Is the number $A$ a multiple of $7$?
- b) Is the number $B$ divisible by $10$? (Hint: see if it has the necessary factors to form $10$).
- c) Is the number $3$ a factor of $A$?
A Physical Education teacher wants to organise their $28$ students into equal groups of more than $2$ people but fewer than $10$, so that no student is left over. Use…
A Physical Education teacher wants to organise their $28$ students into equal groups of more than $2$ people but fewer than $10$, so that no student is left over.
Use the prime factorisation of $28$ to find all possible ways to make the groups.