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.

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.

To factor the number $4200$ into prime factors, Seán and Aoife debate which strategy is more convenient:

• Seán suggests using the traditional vertical bar, starting by dividing systematically by the smallest prime numbers: $2, 2, 2, 3\dots$

• Aoife suggests using a quick factor tree: first she mentally breaks $4200$ down into $42 \times 100$, and then breaks $42$ and $100$ down into smaller branches until reaching primes.

1. a) Will both reach the same final result? What property or theorem of natural numbers guarantees this?
2. b) Compare both methods: which one is faster and carries less risk of error in the divisions for this specific number? Justify your choice.
3. c) Apply Aoife's strategy to write the final prime factorisation of $4200$ in index form.

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 secret natural number $N$ simultaneously satisfies these three conditions:

1. Only the prime factors $2$, $3$, and $5$ appear in its prime factorisation.

2. It is a multiple of $18$ and also a multiple of $20$.

3. It is less than $300$.

1. a) Write the prime factorisation of $18$ and of $20$.
2. b) Thinking of the prime factors as the essential "building blocks" that $N$ must have in order to be divisible by $18$ and by $20$, deduce with reasoning what the minimum exponents of $2$, $3$, and $5$ in the factorisation of $N$ must be.
3. c) Determine what number $N$ is. Could there be a different natural number that satisfies all three conditions? Justify your answer.

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?

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 $m.c.m.$

A Year 8 student claims to have found a number $X$ whose prime factorisation is $2^{2} \times 3^{4} \times 6^{2} \times 5$.

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

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

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

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 Dublin was exactly the time given by their $m.c.m$.

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

1. The prime factorisation 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.

An agricultural cooperative in Wexford is preparing a shipment of $2520\text{ kg}$ of oranges in identical boxes destined for several European markets. To organise the transport, the logistics department requires the following conditions:

1. The weight of each box (in whole kilograms) must be a multiple of $15$, but must be an odd quantity (that is, it cannot contain the prime factor $2$).

2. The total number of boxes prepared must be greater than $20$.

3. The total number of boxes must be a multiple of $8$.

1. a) Calculate the prime factorisation of $2520$.
2. b) Using only the obtained prime factorisation (without trying random divisions), explain which prime factors must necessarily make up the weight of each box and which ones must make up the total number of boxes.
3. c) Deduce with reasoning all the possible combinations of (weight of each box, total number of boxes) that satisfy all transport requirements.

We are looking for the smallest natural number $N$ that satisfies both of these structural conditions at the same time:

• When multiplied by $6$, the result $6 \times N$ is a perfect square (all exponents in its prime factorisation are even numbers).

• When multiplied by $10$, the result $10 \times N$ is a perfect cube (all exponents in its prime factorisation are multiples of $3$).

1. a) Write the prime factorisations of $6$ and $10$, and deduce which prime factors the number $N$ must have as a minimum.
2. b) Assuming that $N = 2^a \times 3^b \times 5^c$, deduce what parity and divisibility conditions the exponents $a$, $b$, and $c$ must necessarily satisfy.
3. c) Find the smallest possible natural values for $a$, $b$, and $c$, express the prime factorisation of $N$, and calculate the numerical value of that number.