Greatest Common Factor

Observe the divisors 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 the common divisors of both numbers.
2. b) Identify which is the largest of these common divisors. What do we call this number?

Calculate the highest common factor of the following numbers using the prime factorization method:

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

Calculate the Highest Common Factor of the following groups of numbers using the prime factorization method:

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 exponent.

Two numbers are called coprime if their only common divisor is $1$. Check if the following pairs of numbers are coprime by calculating their highest common factor:

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

The Euclidean Algorithm is an ancient and effective method for finding the $m.c.d.$ of two numbers through successive divisions. It consists of dividing the larger by the smaller; 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 performing the usual prime factorization afterwards.

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.

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 that exists between the product of two numbers, their $m.c.d.$ and their $m.c.m.$, find out what the Highest Common Factor of these numbers is.

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 $m.c.m.(x, y) = 750$.

Hint: Use the property that relates the product of two numbers to their HCF and their LCM.

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

Ashley has a blue ribbon of $18$ inches and a red ribbon of $24$ inches. She needs to cut them into strips of the same length to make some 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?

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

1. a) How long must each piece be?
2. b) How many crimson pieces will be obtained? And turquoise ones?

In an 8th-grade class, there are $24$ boys and $16$ girls. The Physical Education teacher wants to form teams of the same size, as large as possible, so that in each team there are 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?

For the school's friendly tournament, $48$ 7th-grade students and $60$ 8th-grade students have registered. They want to form teams with the same number of people, without mixing students from different grades and so that the teams are as large as possible.

1. a) How many students will there be in each team?
2. b) How many 7th-grade teams will there be in total? And 8th-grade?

A food bank has received $210$ cans of tuna, $126$ packets of rice and $84$ jars of beans. They want to prepare the largest possible number of identical aid boxes containing all three types of food, so that nothing is left over.

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

We want to decorate a wall in the music classroom that measures $120$ inches high by $180$ inches wide with square cards. If we want the cards to be as large as possible and to cover the wall exactly without having to cut them:

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

A municipal worker wants to tile a rectangular area of a park that measures $24$ feet long by $18$ feet wide. To do this, he is going to use square tiles as large as possible and does not want to have to cut any.

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

An architect is designing the floor of a trophy room for an e-sports club. The room has a rectangular shape $12,6$ feet long by $7,2$ feet wide. They want to cover the floor with square tiles of the largest possible size, so that none have to be cut and the design is perfectly symmetrical.

However, the architect wants to leave a small ventilation channel $60$ inches 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 this inner rectangle and how many tiles will be needed in total.

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

They want to cut them into pieces of the same length, with nothing left over, to distribute them among different teams of volunteers. 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 $15$ feet.

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

Imagine you have a plot of land in the shape of a scalene triangle whose sides measure $210$, $270$ and $330$ feet. You want to place wooden posts around the entire perimeter at equal distances, so that there is a post at each vertex and the distance between posts is the maximum possible.

If each post costs $\$14,50$ and the labor to place each one is $\$6$, what will be the total budget to fence the plot?

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 in 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 only one type of goods and there can be no leftover space in any of them, what must be the volume of each cubic container? How many textile containers will there be in the warehouse?

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.

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 divisors.