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 Sierra Madre, the instructors have two climbing ropes: one measures $20$ meters and the other $28$ meters. 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 an 8th grade 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 Baja California has produced $45$ liters of virgin olive oil and $60$ liters 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 $20$

María 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 color?

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 municipal worker wants to tile a rectangular area of a park measuring $24$ meters long by $18$ meters 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?

An architect is designing the floor of an e-sports club trophy room. The room is rectangular, measuring $12.6$ meters long by $7.2$ meters 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 $x \times y = 11\,250$ 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 $a = 24 \times p$ and $b = 24 \times q$, where $p$ and $q$ must be coprime.

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

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 center sets the condition that the pieces must be as long as possible, but for safety reasons, no piece can measure more than $15$ meters.

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 $330$ meters. 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 $290$ pesos and the labor to install each one is $120$ pesos, what will be the total budget to fence the land?