Exercises: Highest common factor

Calculate all the divisors of the numbers $12$ and $18$. Then, indicate which are the common divisors and point out which of them is the Greatest Common Factor (GCF).

Ashley has $20$ star stickers and $30$ planet stickers. She wants to make equal envelopes with the largest possible number of stickers of each type, without any left over. How many stickers of each kind must she put in each envelope? (Hint: Look for the greatest common divisor of $20$ and $30$).

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

• $A = 2^{2} \cdot 3 \cdot 5$

• $B = 2 \cdot 3^{2}$

Remember the pedagogical rule: "For the GCF, we take the common factors raised to the lowest exponent".

A gardener wants to cut two hoses, one of $15$ feet and another of $25$ feet, into pieces that measure the same and are as long as possible, so that no hose is wasted. What length will each piece be?

Work out the $MCD$ of the following pairs of numbers mentally. They are small numbers where you can identify the greatest 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?

In the technology workshop, the students have two wooden slats of $36$ inches and $48$ inches in length. They need to cut them into pieces of the same length, this 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 slats?

We say that two numbers are relatively prime when their only common divisor is $1$. Check if the numbers $8$ and $15$ are relatively prime by finding their $GCF$.

In a library they want to organize $45$ adventure books and $60$ science fiction books on shelves, so that on each shelf there is the same number of books and that this 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 ones?

Calculate the Greatest Common Factor ($M.C.D.$) of the following groups of numbers using the prime factorization method:

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

An electrician has two rolls of leftover cable from a construction site: a blue one that measures $120$ feet and a black one of $96$ feet. He wants to cut them into pieces of the same length, so that they are as long as possible and there is no cable left over on either roll.

1. a) How long must each piece be?
2. b) How many pieces will he get in total between the two colors?

A team-building day has been organized at the middle school. There are $54$ students in 6th grade and $42$ students in 7th grade. They want to form teams (all from the same grade or mixed, as decided by the teacher) but on the condition that all the teams in the school have exactly the same number of people and that this number is the maximum possible without anyone being left out.

1. a) How many students will there be in each team?
2. b) How many 6th grade teams and how many 7th grade teams will be formed?

Look at the following prime factorizations of three numbers and, without carrying out the multiplications, indicate what their $\text{HCF}$ is:

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

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

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

A craft shop wants to cut a sheet of foam board $150$ inches long by $90$ inches wide into equal squares, so that they are as large as possible and no piece is left over.

1. a) What dimensions will the side of each square have?
2. b) If each square is sold for $\$2,50$, how much money will they get from selling all the squares that come out of the sheet?

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

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

For a food bank, a supermarket donates $36$ cans of beans, $48$ cartons of milk and $60$ packages of rice. They want to prepare the largest possible number of equal bags to distribute, so that in each bag there is the same number of products of each type.

1. a) How many bags will they be able to prepare at most?
2. b) What will each bag contain?

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

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

Explain your answer by showing the prime factorization of the chosen numbers.

A food bank has received three donations of basic products: $540$ gallons of milk, $420$ pounds of rice and $360$ packages of pasta. They want to organize identical batches to distribute among the largest possible number of families, so that in each batch there is the same amount of each product and nothing is left over.

1. a) How many families will they be able to help at most?
2. b) What amount of each product will each of these families receive?

An urban artist wants to design a mural on a rectangular wall that measures $210$ inches high by $165$ inches wide. For her design, she wants to divide the entire wall into equal squares, so that they are as large as possible and she does not have to cut any pieces.

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

In a technology workshop, the students have three wooden slats of $120$ inches, $144$ inches and $180$ inches in length. The teacher asks them to cut them into equal pieces, with no wood left over, and of the greatest possible length to make bridge structures.

1. a) What length must each piece be?
2. b) How many pieces will they get in total by adding the three slats?

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. Explain the steps you follow to find it using the concept of Greatest Common Factor.

An organic farmer has picked $250$ red apples, $325$ green apples and $175$ yellow apples. He wants to pack them into equal boxes (with the same number of apples of a single color per box) to send them to different shops. If he wants the boxes to contain the maximum possible number of pieces:

1. a) How many apples will there be in each box?
2. b) How many boxes of each type of apple will he send?

In a chemistry laboratory they have three containers with different solutions: one with $135$ fl oz, another with $180$ fl oz and a third with $225$ fl oz. They want to distribute all the liquid into identical test tubes, with the greatest possible capacity, so that all the tubes are completely full and the substances are not mixed.

1. a) What will be the maximum capacity of each test tube?
2. b) How many tubes will be filled from the container with the largest volume?

We know that the Greatest Common Factor 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) Explain why it could not be the number $60$ using prime factorization.

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

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