Prime Factorization
At Doña Altagracia's corner store, $24$ bottles of juice just arrived. They can be arranged in many ways: $2 \times 12$, $3 \times 8$, $4 \times 6$... But there is only one way to write $24$ using only prime factors: it is its mathematical "fingerprint." What is it? By the end of this section you will have it.
Fundamental theorem: a unique factorization
Go back to the start: the $24$ bottles can be summed up as $24 = 2^3 \cdot 3$.
State which number corresponds to each of the following prime factorisations: a) $2^{2} \times 3 \times 5$ b) $2^{3} \times 7$ c) $3^{2} \times 11$
State which number corresponds to each of the following prime factorisations:
- a) $2^{2} \times 3 \times 5$
- b) $2^{3} \times 7$
- c) $3^{2} \times 11$
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 factor: you keep splitting the number into two factors, any two, 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.
Copy this prime factorisation diagram into your notebook and fill in the blanks (marked with the letter $x$) with the corresponding numbers:
Copy this prime factorisation diagram into your notebook and fill in the blanks (marked with the letter $x$) with the corresponding numbers:
The vertical bar technique
The fastest 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) $80$
Find the prime factorisation of the numbers $720$ and $1260$ using the division ladder method. Once you have obtained the expressions in index form, state which prime…
Find the prime factorisation of the numbers $720$ and $1260$ using the division ladder method. Once you have obtained the expressions in index form, state which prime factors they have in common and the highest exponent with which each appears in both factorisations.
Using the factorization to reason
Once you have factored a number, you can answer questions about multiples and divisors without calculating its value.
Listing all the divisors
Every divisor of $N$ is formed by taking each prime with an exponent that goes from $0$ up to the one it has in $N$, and multiplying. With a table you won't miss any.
So, if Wilkin wants to split $28$ students into equal groups of more than $2$ and fewer than $10$ people, from the list of divisors only $4$ and $7$ work.
Look at the prime factorisation of the following numbers: $A = 2^{2} \times 3 \times 7$ $B = 2 \times 3^{2} \times 5$ Without carrying out the multiplications, answer…
Look at the prime factorisation of the following numbers:
$A = 2^{2} \times 3 \times 7$
$B = 2 \times 3^{2} \times 5$
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.
Using the prime factorisation of the number $150$, answer the following questions, giving reasons for your answers: a) Is $150$ divisible by $10$? b) Is $150$ divisible…
Using the prime factorisation of the number $150$, answer the following questions, giving reasons for your answers:
- a) Is $150$ divisible by $10$?
- b) Is $150$ divisible by $4$?
- c) Is $150$ divisible by $15$?
More uses: number of divisors and perfect squares
Counting divisors
Perfect squares
A number is a perfect square if all the exponents in its factorization are even. For example, $36 = 2^2\cdot3^2$ and $\sqrt{36}=2\cdot3=6$. To turn a number into a perfect square, you multiply by the primes that have an odd exponent. Similarly, for a cube you need exponents that are multiples of $3$ (the edge of a cube comes from grouping the factors into three equal groups).
GCF and LCM using the factorization
Find the number whose prime factorisation is $2^3 \times 3^2 \times 11$. Once found, explain whether that number is a multiple of $6$ and $22$ looking only at its prime…
Find the number whose prime factorisation is $2^3 \times 3^2 \times 11$. Once found, explain whether that number is a multiple of $6$ and $22$ looking only at its prime factors (without carrying out division).
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.
We say a number is a perfect square if all exponents in its prime factorisation are even numbers. a) Find the prime factorisation of the number $180$. Is it a perfect…
We say a number is a perfect square if all exponents in its prime factorisation are even numbers.
- a) Find the prime factorisation of the number $180$. Is it a perfect square?
- b) What is the smallest number you would need to multiply $180$ by so that the result is a perfect square? Justify your answer by analysing the exponents.
Consider two natural numbers, $A$ and $B$, whose prime factorizations are as follows: $A = 2^{3} \times 3^{x} \times 5^{2}$ $B = 2^{y} \times 3^{3} \times 7^{1}$ Knowing…
Consider two natural numbers, $A$ and $B$, whose prime factorizations are as follows:
$A = 2^{3} \times 3^{x} \times 5^{2}$
$B = 2^{y} \times 3^{3} \times 7^{1}$
Knowing that their greatest common divisor is $M.C.D.(A, B) = 36$, find the value of the unknowns $x$ and $y$. Once found, calculate the value of the least common multiple $m.c.m.(A, B)$, also expressed in its prime factorization form.