Prime factorisation

Classify the following natural numbers into prime or composite. Remember that prime numbers are those that have only two factors: $1$ and themselves.

$14, 17, 21, 23, 27, 29, 33, 37$

State which number corresponds to each of the following prime factorisations:

1. a) $2^{2} \times 3 \times 5$

1. b) $2^{3} \times 7$

1. c) $3^{2} \times 11$

What is the largest two-digit number that has only $2$ and $3$ as prime factors in its prime factorisation?

Helpful example: the number $6 = 2 \times 3$ satisfies the condition, but it is not the largest two-digit 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.

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:

Consider the number obtained by multiplying $A = 24 \times 35 \times 55$.

1. a) Estimate: Before calculating the product and without doing long divisions, make a mental estimate of how many prime factors in total (counting repetitions; for example, in $2^3$ there are $3$ prime factors) the number $A$ will have. Briefly explain what you look at mentally to make this estimate.
2. b) Calculate: Find the prime factorisations of $24$, $35$, and $55$ separately. From them, write the complete prime factorisation of $A$ by grouping the factors into powers.
3. c) Check and reflect: Add the exponents of your factorisation to find the exact number of prime factors of $A$ and compare it with your estimate. Why is it much more advantageous to factor the three numbers separately instead of first calculating the multiplication $24 \times 35 \times 55 = 46\,200$ and then factoring it?

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:

1. a) $36$

1. b) $45$

1. c) $80$

Cian has to factor the number $540$ into prime factors and uses the vertical bar method, writing in his notebook:

$540 \mid 10$

$\phantom{0}54 \mid 9$

$\phantom{00}6 \mid 6$

$\phantom{00}1$

When he finishes, he claims: "Since $10 \times 9 \times 6 = 540$, the prime factorisation of $540$ is $10 \times 9 \times 6$ and it took me half the time".

1. a) Explain what conceptual error Cian makes when applying the definition of prime factorisation.
2. b) Why is it not valid to use numbers like $10$, $9$, or $6$ in the right-hand column of the vertical bar method?
3. c) Correct Cian's exercise: find the correct prime factorisation of $540$ expressed as a product of powers of prime numbers.

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:

1. a) Is the number $A$ a multiple of $7$?
2. b) Is the number $B$ divisible by $10$? (Hint: see if it has the necessary factors to form $10$).
3. 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 the prime factorisation of $28$ to find all possible ways to make the groups.

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 $360$.

1. a) Find the prime factorisation of $360$.
2. 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.

1. a) Find the prime factorisation of the number $180$. Is it a perfect square?
2. 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 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.