Highest 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 $MCD$:

• $A = 2^{2} \cdot 3 \cdot 5$

• $B = 2 \cdot 3^{2}$

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 $\text{HCF}$:

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

$B = 2^2 \cdot 3^3$

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

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 $25$

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?

At the school, a day of group activities has been organized. There are $54$ students in 1st ESO and $42$ students in 2nd ESO. The students are to be split into teams so that each team consists only of students from the same year (students from 1st and 2nd year are not mixed). All teams must have exactly the same number of people, and that number must be as large as possible without anyone being left over.

1. a) How many students will there be in each team?

1. b) How many teams of 1st ESO and how many teams of 2nd ESO 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?