Exercises: 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.

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

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) $80$

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) $3^{2} \times 11$

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.

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.

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.

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.

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

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.

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 $15$?

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.

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.

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.

A hacker has blocked access to a server and left the following clue about the numerical password, which is a four-digit number $N$:

• The prime factorisation of $N$ is of the form $2^{a} \times 3^{b} \times 5^{c}$.

• The total number of factors of $N$ is exactly $30$.

• The number is a multiple of $100$ but is not a multiple of $8$.

Determine with reasoning the value of the password $N$ and show its full prime factorisation.

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.

A logistics company uses small cubical boxes to store parts. The total volume of a cargo container is $151200$ $cm^{3}$. The container must be completely filled with rectangular packages measuring $a$ cm wide, $b$ cm long, and $c$ cm high, where $a$, $b$, and $c$ are distinct prime numbers.

Given that the container can hold an exact number of packages that is a multiple of $10$:

1. Find the prime factorisation of $151200$.

2. Identify the possible dimensions of the packages if the volume of each package must be greater than $100$ $cm^{3}$.

We have three natural numbers decomposed into prime factors:

$n_1 = 2^{3} \times p \times 7$

$n_2 = 2^{q} \times 3^{2} \times 5$

$n_3 = 3 \times 5^{2} \times 11$

If we know that:

• $M.C.D.(n_1, n_2) = 12$

• $m.c.m.(n_2, n_3) = 2^{2} \times 3^{2} \times 5^{2} \times 11$

Find the values of the unknown prime numbers or exponents $p$ and $q$. Justify your answer using the properties of prime factorization for calculating the $M.C.D.$ and the $m.c.m.$

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.

In a satellite network, Satellite A completes an orbit every $1260$ minutes, Satellite B every $1584$ minutes, and Satellite C every $n$ minutes.

It is known that the last time all three coincided above Santo Domingo was exactly the time given by their $m.c.m$.

If the prime factorization of $n$ is $2^{3} \times 3^{2} \times 7$, calculate:

1. The prime factorization of the times for satellites A and B.

2. The total time (in minutes) that must pass before the three satellites coincide again. Leave your answer expressed as a product of powers of prime factors.

Find the smallest number that simultaneously satisfies these three conditions:

• It is a perfect square (all exponents in its prime factorisation are even).

• It is a multiple of $42$.

• It is a multiple of $60$.

To solve this, use the prime factorisation of $42$ and $60$ and reason which factors and exponents you must add to obtain the smallest perfect square.

You want to design a special barcode based on a number $K$. This number is formed by multiplying the first $n$ prime numbers consecutively (for example, if $n=3$, the number would be $2 \times 3 \times 5 = 30$).

If we want the number $K$ to:

1. Be greater than $1000$ but less than $5000$.

2. Have exactly $5$ distinct prime factors in its prime factorization.

What is the value of $K$? Show the list of prime numbers you used and check that the result is within the required range.