Greatest Common Factor

Find all the divisors of the numbers $12$ and $18$. Then, state which are the common divisors and point out which of them is the Greatest Common Divisor (GCD).

You can use a diagram like the one in the figure: write the divisors of each number in each circle and, in the area where they overlap, the ones that are common.

Work out in your head the $MCD$ of the following pairs of numbers. These are small numbers where you can identify the highest common factor at a glance:

1. a) $MCD(4, 8)$
2. b) $MCD(5, 10)$
3. c) $MCD(7, 14)$
4. d) $MCD(10, 20)$

What do all these results have in common?

Look at the prime factorisations of these two numbers and find their $MCD$:

• $A = 2^{2} \times 3 \times 5$

• $B = 2 \times 3^{2}$

Remember the rule: "For the HCF, we take the common factors raised to the lowest power".

Calculate the Highest Common Factor ($M.C.D.$) of the following sets of numbers using prime factorisation:

1. a) $48$ and $72$
2. b) $30, 45$ and $60$
3. c) $120$ and $150$

Look at the following prime factorisations of three numbers and, without performing the multiplications, state their $\text{HCF}$:

$A = 2^3 \times 3^2 \times 5$

$B = 2^2 \times 3^3$

$C = 2^2 \times 3^2 \times 7$

Two numbers are said to be coprime if their only common factor is $1$. Check whether the following pairs of numbers are coprime by calculating their $M.C.D.$:

1. a) $14$ and $15$
2. b) $22$ and $33$
3. c) $21$ and $25$

Make up two distinct natural numbers, $a$ and $b$, such that the following condition is met:

$M.C.D.(a, b) = 12$

Reason your answer by showing the prime factorisation of the chosen numbers.

In the technology workshop, the students have two wooden battens measuring $36$ cm and $48$ cm in length. They need to cut them into pieces of the same length, with this length being the maximum possible.

1. a) What will be the length of each piece?
2. b) How many pieces will they get in total from the two battens?

In a library, they want to organise $45$ adventure books and $60$ science fiction books on shelves so that each shelf has the same number of books and this number is the maximum possible.

1. a) How many books will they put on each shelf?
2. b) How many shelves will they use for the adventure books and how many for the science fiction books?

A community day has been organized at the school. There are $54$ 7th grade students and $42$ 8th grade students. They want to form teams (all from the same class or mixed, as the homeroom teacher decides) on condition that all teams in the school have exactly the same number of people and that this number is as large as possible without anyone being left over.

1. a) How many students will be in each team?
2. b) How many 7th grade teams and how many 8th grade teams will be formed?

For a food bank, a supermarket donates $36$ tins of pulses, $48$ cartons of milk and $60$ bags of rice. They want to prepare the maximum possible number of identical bags to distribute, such that each bag contains the same number of items of each type.

1. a) What is the maximum number of bags they can prepare?
2. b) What will each bag contain?

A street artist wants to design a mural on a rectangular wall measuring $210$ cm high by $165$ cm wide. For her design, she wants to divide the whole wall into equal squares, such that these are as large as possible and she does not have to cut any pieces.

1. a) What must be the length of the side of each square to cover the entire wall?
2. b) How many squares will the mural have in total?

A gardener wants to surround a rectangular vegetable patch $90$ meters long and $126$ meters wide with fence posts. The owner of the patch has asked him to place a post at each corner and for the distance between consecutive posts to always be the same and as large as possible.

1. a) What distance should he leave between each post?
2. b) How many posts will he need in total to surround the entire perimeter of the vegetable patch?

We are looking for the largest possible number that, when dividing the numbers $159$ and $195$, leaves a remainder of $3$ units in both cases. Reason through the steps you follow to find it using the concept of Highest Common Factor.

We know that the Greatest Common Divisor of two numbers is $12$ and that one of those numbers is $48$. If the other number is greater than $50$ but less than $70$:

1. a) Find what that second number is.
2. b) Justify your answer using prime factorisation.