Highest common factor

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?

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?

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 $25$

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.

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.

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.

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?

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?

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.