Prime Factorization

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) $3^{2} \times 11$

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

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

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.

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.

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) $80$

Copy this prime factorisation diagram into your notebook and fill in the blanks (marked with the letter $x$) with the corresponding 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.

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).

Using the prime factorisation of the number $150$, answer the following questions, giving reasons for your answers:

1. a) Is $150$ divisible by $10$?
2. b) Is $150$ divisible by $4$?
3. c) Is $150$ divisible by $15$?

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.

The volume of a cube-shaped box is exactly $1728$ cm$^3$.

1. a) Find the prime factorisation of the number $1728$.
2. b) Knowing that the volume of a cube is calculated as $V = l^3$ (where $l$ is the edge length), use the factorisation above to group the factors into three identical sets and find the length of the edge.

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.

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.

A professional Instagram photographer wants to organise his photo feed. He has $210$ images and wants to post them in equal blocks.

1. a) Find the prime factorisation of $210$.
2. b) Write down four different ways to group the photos (for example: "3 blocks of 70 photos") using combinations of its prime factors.

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.

An 8th grade student claims to have found a number $X$ whose prime factorization is $2^{2} \times 3^{4} \times 6^{2} \times 5$.

1. Explain why this expression is not a valid prime factorization.

2. Correct the expression so that it is a true prime factorization.

3. Without calculating the numerical value of $X$, determine how many factors it has. Show your step-by-step work.