Exercises: Highest common factor

Calculate the greatest common divisor (g.c.d.) of the following numbers using the prime factorization method (remember: you take the common factors raised to the lowest exponent):

1. a) $12$ and $18$
2. b) $20$ and $30$
3. c) $15$ and $25$

Look at the factors of the numbers $24$ and $36$:

• $Div(24) = \{1, 2, 3, 4, 6, 8, 12, 24\}$

• $Div(36) = \{1, 2, 3, 4, 6, 9, 12, 18, 36\}$

From these lists:

1. a) Write down the common factors of both numbers.
2. b) Identify the highest of these common factors. What do we call this number?

At an adventure camp in the Pyrenees, the instructors have two climbing ropes: one measures $20$ metres and the other $28$ metres. They want to cut them into equal pieces of the maximum possible length, with no rope left over. How long should each piece be?

In a Year 8 class, there are $24$ male students and $16$ female students. The PE teacher wants to form equal-sized teams, as large as possible, so that each team contains only boys or only girls.

1. a) How many students will there be in each team?
2. b) How many boys' teams and how many girls' teams will be formed?

A farmer from Jaén has produced $45$ litres of virgin olive oil and $60$ litres of sunflower oil. He wants to bottle them in identical containers with the maximum possible capacity without mixing the two types of oil. What capacity should each container have?

We want to decorate a wall in the music room measuring $120$ cm high by $180$ cm wide using square pieces of card. If we want the pieces of card to be as large as possible and cover the wall exactly without having to cut them:

1. a) What length should each side of the pieces of card be?
2. b) How many pieces of card will we need in total?

Two numbers are called co-prime if their only common factor is $1$. Check whether the following pairs of numbers are co-prime by calculating their highest common factor:

1. a) $8$ and $15$
2. b) $14$ and $21$
3. c) $9$ and $20$

Ainhoa has a blue ribbon measuring $18$ cm and a red ribbon measuring $24$ cm. She needs to cut them into strips of equal length to make decorative bows. If she wants the strips to be as long as possible and does not want to waste any ribbon:

1. a) What will be the length of each strip?
2. b) How many strips will she get of each colour?

Marcos has to calculate the greatest common divisor of $72$ and $120$. After decomposing both numbers into prime factors, he writes in his notebook:

And he concludes: "Since the gcd is the greatest, I choose all the factors that appear raised to their highest exponent: $\text{gcd}(72, 120) = 2^3 \cdot 3^2 \cdot 5 = 360$".

1. a) Without doing complex calculations, explain why Marcos's result is impossible by reasoning about the meaning of the word "divisor".
2. b) What concept has Marcos actually calculated instead of the gcd?
3. c) Explain the correct rule for calculating the gcd from the prime factorizations and find the correct result.

You are going to calculate the greatest common divisor of $48$ and $80$, but before applying any algorithm you must anticipate what the solution will look like:

1. a) Estimate: Without decomposing the numbers into prime factors, answer with reasoning:

• Can the result be greater than $48$? Why?

• Knowing that both $48$ and $80$ are even, could the result be an odd number?

• Check mentally whether $10$ could be a common divisor of both. Why is it immediately ruled out?

1. b) Check: Decompose $48$ and $80$ into prime factors, calculate the $\text{gcd}(48, 80)$, and verify whether the value obtained satisfies all the deductions you made in your estimation.

Find the Highest Common Factor of the following groups of numbers using prime factorisation:

1. a) $m.c.d.(120, 150)$
2. b) $m.c.d.(72, 108, 144)$
3. c) $m.c.d.(45, 60, 105)$

Remember to first express each number as a product of its prime factors and then select the common factors raised to the lowest power.

A craft shop has two silk ribbons, a crimson one that is $160$ cm long and a turquoise one that is $120$ cm long. They want to cut both ribbons into pieces of the same length, so that the pieces are as long as possible and no fabric is left over.

1. a) How long must each piece be?

1. b) How many crimson pieces will you get? And how many turquoise pieces?

For the school sports day tournament, $48$ 1st ESO students and $60$ 2nd ESO students have signed up. Teams are to be formed with the same number of people, without mixing students from different year groups, making the teams as large as possible.

1. a) How many students will there be in each team?
2. b) How many 1st ESO teams will there be in total? And 2nd ESO?

A council worker wants to tile a rectangular area of a park measuring $24$ metres long by $18$ metres wide. To do this, they will use square tiles that are as large as possible and do not want to cut any.

1. a) Determine the side length of each tile.
2. b) Calculate the total number of tiles needed to cover the entire surface.

A food bank has received $210$ tins of tuna, $126$ packets of rice and $84$ jars of pulses. They want to prepare the maximum possible number of identical aid boxes containing all three types of food, with nothing left over.

1. a) How many aid boxes will they be able to prepare?
2. b) How many items of each type will there be in each box?

The Euclidean algorithm is an ancient and effective method for finding the $m.c.d.$ of two numbers using successive division. It consists of dividing the larger number by the smaller one; if the remainder is not zero, the divisor is divided by the remainder, and so on until the remainder is zero. The last divisor used is the $m.c.d.$

Use this method to find the $m.c.d.(525, 225)$. Check the result by doing the usual prime factorisation afterwards.

It is known that the product of two natural numbers is $2160$ and that their lowest common multiple ($m.c.m.$) is $180$. Using the relationship between the product of two numbers, their $m.c.d.$ and their $m.c.m.$, find out what the Highest Common Factor of those numbers is.

A comic collector has three shelves with the following quantities: $135$ superhero comics, $90$ manga and $60$ European comics. He wants to store them in identical storage boxes with the same number of copies in each, without mixing them and in such a way that the number of storage boxes is as small as possible (that is, each storage box is as full as possible).

1. a) How many comics should he put in each storage box?
2. b) How many storage boxes will he need in total for his whole collection?

To calculate the greatest common divisor of two numbers, you have different strategies available:

• Method A (Direct divisibility criterion): If the larger number is divisible by the smaller one, the smaller one is directly the gcd.

• Method B (Exhaustive list of divisors): Write down all the divisors of both numbers and look for the greatest one in common.

• Method C (Prime factorization): Decompose into prime factors and select the common ones with the lowest exponent.

Analyze the following two pairs of numbers:

1) $28$ and $112$

2) $90$ and $126$

For each of the two pairs:

1. a) Choose with reasoning which of the three methods is the fastest and most efficient, justifying why you rule out the other two in that specific case.
2. b) Apply the chosen method to calculate the gcd of each pair.

An olive oil cooperative in Jaén wants to prepare identical promotional packs for a food fair. It has three bottled varieties: $168$ bottles of picual oil, $252$ bottles of arbequina oil, and $336$ bottles of hojiblanca oil.

They wish to pack all the oil into identical boxes, so that each box contains exactly the same number of bottles of each variety, without mixing bottles unevenly and without leaving any spare bottles. Furthermore, to make the packaging cost-effective, they want to prepare the greatest possible number of boxes.

1. a) Translate the situation into mathematical language: what concept must you calculate to find out the maximum number of boxes that can be assembled? Justify your answer.
2. b) Calculate how many promotional boxes can be prepared in total.
3. c) Determine how many bottles of each oil variety each individual box will contain.

An architect is designing the floor of an e-sports club trophy room. The room is rectangular, measuring $12,6$ metres long by $7,2$ metres wide. The intention is to cover the floor with square tiles of the largest possible size, so that none need to be cut and the design is perfectly symmetrical.

However, the architect wants to leave a small ventilation channel $60$ cm wide along the entire perimeter of the room, so the actual area to be tiled will be a smaller inner rectangle.

Calculate the side length of the largest tile that can be used to cover that inner rectangle and how many tiles will be needed in total.

Find two natural numbers, which we will call $x$ and $y$, knowing that their product is $x \cdot y = 11\,250$ and that their lowest common multiple is $m.c.m.(x, y) = 750$.

A logistics company wants to transport three types of goods in identical cubic containers, as large as possible, to make the most of the space in a large cargo warehouse. The quantities of goods are measured by volume:

• Technology boxes: $1\,512$ $m^3$

• Food boxes: $1\,848$ $m^3$

• Textile boxes: $1\,260$ $m^3$

If each container must be filled with a single type of good and no space can be left over in any of them, what must the volume of each cubic container be? How many textile containers will there be in the warehouse?

Use the Euclidean algorithm to find the $m.c.d.(2\,310, 1\,485)$. Once found, state whether the numbers are coprime and justify your answer based on the divisors obtained during the algorithm process.

Determine all pairs of natural numbers whose sum is $168$ and whose highest common factor is $m.c.d.(a, b) = 24$.

Help: If the HCF is 24, you can write the numbers as $a = 24 \cdot p$ and $b = 24 \cdot q$, where $p$ and $q$ must be coprime.

A marine wildlife rescue centre has received a donation of three heavy-duty hoses to clean the tanks. The lengths of the hoses are $156$ metres, $144$ metres and $108$ metres.

They want to cut them into pieces of equal length, with nothing left over, to distribute them among different volunteer teams. The head of the centre sets the condition that the pieces must be as long as possible, but for safety reasons, no piece can measure more than $15$ metres.

What length should each piece be to meet all conditions? How many volunteers will receive a hose?

We have two natural numbers $A$ and $B$ such that their $m.c.d.(A, B) = 15$. If we multiply the number $A$ by $12$ and the number $B$ by $12$, what will be the new highest common factor of the resulting numbers? Justify your answer by applying the properties of factors.

Imagine you have a plot of land in the shape of a scalene triangle whose sides measure $210$, $270$ and $330$ metres. Wooden posts are to be placed along the entire perimeter at equal distances, so that there is a post at each vertex and the distance between posts is as large as possible.

If each post costs $14,50$ € and the labour to install each one is $6$ €, what will be the total budget to fence the land?

We are looking for two three-digit natural numbers, $a$ and $b$ (with $a < b$), whose initial values are unknown, but we know that they simultaneously satisfy two conditions:

1. a) Apply the property relating the product of two numbers to their gcd and their lcm to calculate the exact value of the product $a \cdot b$.
2. b) If we write $a = 18 \cdot x$ and $b = 18 \cdot y$, deduce what value the product $x \cdot y$ must have and reason why $x$ and $y$ must necessarily be coprime (that is, they cannot share any common factor).
3. c) Decompose the product $x \cdot y$ into prime factors, analyze all possible pairs of factors $(x, y)$ that are coprime, and deduce which are the only values of $a$ and $b$ that satisfy that both have exactly three digits.

A parcel logistics warehouse in Zaragoza is preparing shipments for three Aragonese regions. They have three pallets with different amounts of school supply parcels: the first contains $540$ parcels, the second $720$, and the third $900$. They want to repack them into smaller containers that each carry exactly the same number of parcels, without mixing material from different pallets and without leaving any parcels leftover.

However, the transport company imposes two restrictions for ergonomic safety reasons:

• Each container must carry at least $20$ parcels.

• No container may carry more than $50$ parcels.

1. a) Calculate the $\text{gcd}(540, 720, 900)$ and explain why this result alone does not directly solve the problem.
2. b) Reason why any valid container size that allows distributing the parcels without leaving any leftover must necessarily be a divisor of the obtained gcd.
3. c) Find all possible container sizes (number of parcels per container) that meet the restrictions imposed by the transport company.
4. d) Which size is best to choose if the goal is to use the smallest total number of containers? Calculate how many containers will be needed in total with that choice.

Unit challenge. "Home energy audit". Become an energy auditor for your home using real data obtained from your latest electricity bill or the official website of Red Eléctrica de España (REE). Prepare a brief report following these stages:

1. Data and scientific notation: Check your monthly consumption in kWh ($1\text{ kWh} = 3,6 \cdot 10^6\text{ J}$) and convert it to joules using scientific notation and powers. Record the price per kWh with decimals (for example, $0,15\text{ \euro}$) and calculate the base cost.

2. Divisibility and measurements: Select two recurring consumption routines (such as running the washing machine every $3$ days or the dishwasher every $4$) and use prime factorisation and the $\text{l.c.m.}$ to predict when they will coincide. In addition, if you intend to insulate a square area of your floor of $12,25\text{ m}^2$, find the length of the side by applying the square root.

3. Equation and balance sheet: Set up and solve a linear equation to find out how many hours you can keep a radiator on without exceeding a spending limit of $45\text{ \euro}$. Design a monthly balance sheet in a spreadsheet using integers (positive savings and negative costs).

The following will be assessed:

• Rigour in the use of powers, scientific notation, and operations with decimals.

• Correct application of divisibility ($\text{l.c.m.}$) and square roots.

• Algebraic formulation and valid resolution of the equation.

• Clear organisation of data in the spreadsheet and reasoned conclusions.