Exercises: Multiples and factors

In a video game shop, they want to organise a shelf with $30$ games such that all rows have the same number of games and none are left over. Use the divisibility relation to complete the following sentences:

1. a) If they put $5$ games in each row, there will be $6$ rows. Therefore, $30$ is a ________ of $5$.
2. b) Since the division $30 : 10$ has no remainder, we say that $10$ is a ________ of $30$.
3. c) The number $30$ is ________ by $2, 3$ and $5$ at the same time.

Look at the following representation of "rectangular numbers" for the number $12$. Each rectangle represents a way of multiplying two numbers to get $12$.

Based on the idea that the factors are the side lengths of these rectangles, write down all the factors of the number $18$ by drawing or imagining its possible rectangular shapes.

A content creator likes the duration of their short videos to be a multiple of $9$. If they want to upload a video lasting between $40$ and $70$ seconds, what are all the possible durations (in seconds) they can choose?

Classify the following numbers from the list as prime or composite. Remember that composite numbers can be "broken down" as the product of other smaller numbers:

Explain why you chose the number $21$ as composite, stating which numbers it can be divided by besides $1$ and itself.

Apply the divisibility tests to state whether the number $4,350$ is divisible by the following values (answer YES/NO and briefly justify):

1. a) Is it divisible by $2$?
2. b) Is it divisible by $3$?
3. c) Is it divisible by $5$?
4. d) Is it divisible by $10$?

Complete the following "factor tree" to find the prime factorisation of the number $60$.

Once completed, write the final factorisation using powers: $60 = 2^{\dots} \times \dots \times \dots$

Two alarm clocks go off at different intervals. The first goes off every $15$ minutes and the second every $20$ minutes. If they have just gone off together right now at $08:00$ in the morning:

1. a) In how many minutes will they sound together again for the first time? (Calculate the $m.c.m.$).
2. b) At what exact time will they coincide again?

Aoife has $24$ red stickers and $36$ blue stickers. She wants to make identical surprise envelopes for her friends, such that each envelope has the same number of stickers of each colour and none are left over.

1. a) What is the maximum number of envelopes she can prepare? (Calculate the $HCF$.).
2. b) How many stickers of each colour will be in each of those envelopes?

In a Mathematics exam on divisibility, Cian makes two common mistakes in his reasoning:

1. He claims that "the number $91$ is prime because it is not even, it does not end in $0$ or $5$, and the sum of its digits ($9+1=10$) is not a multiple of $3$".

2. He claims that "to check whether the number $3,514$ is divisible by $4$, it is enough to see if its last digit ($4$) is divisible by $4$, so $3,514$ is indeed divisible by $4$".

Identify the reasoning flaw in each of Cian's two statements, explain why they are incorrect, and write the appropriate correction for each case.

We want to decipher a three-digit numerical code of the form $4AB$ (where $A$ represents the tens and $B$ the units). The following clues about its divisibility are known:

• The number is divisible by $5$, but is not divisible by $10$.

• The number is a multiple of $3$, but is not a multiple of $9$.

Find the exact value of the units digit $B$ and determine, with reasoning, all possible values that the digit $A$ can take so that both conditions are met simultaneously.

A tech shop has received a shipment of tablets. The manager knows that the total number of tablets is a four-digit number of the form $4A7B$. To organise the inventory, the number needs to satisfy the following conditions:

• It must be divisible by $5$ and by $2$ at the same time.

• It must be a multiple of $9$.

Determine all possible values of $A$ and $B$ so that the manager can organise the inventory.

Natural numbers can be represented using rectangles based on their factors. For example, the number $12$ can be represented as a $3 \times 4$ rectangle or a $2 \times 6$ rectangle.

Look at the following diagram and determine which number it is, knowing that it is the smallest number that can form exactly four different rectangles, one of which is a square as in the diagram (considering that a square is a special type of rectangle and that a base of $1 \times n$ also counts).

Three content creators on a streaming platform upload videos at different frequencies:

• The first uploads a video every $18$ hours.

• The second uploads a video every $24$ hours.

• The third uploads a video every $2$ days.

If all three coincided by uploading a video today, Monday, at $09:00$ in the morning, on what day and at what time will all three coincide again for the first time?

An esports club wants to organise a charity tournament. They have $120$ blue t-shirts, $180$ red t-shirts and $210$ white t-shirts. They want to prepare gift backpacks containing the same number of t-shirts of each colour, distributing all the t-shirts they have and without mixing colours within each individual package, while maximising the number of t-shirts per backpack.

1. a) What is the maximum number of t-shirts that can be in each backpack?
2. b) How many backpacks of each colour will be obtained?

Without calculating the numerical value of the number $N$, whose prime factorisation is $N = 2^3 \times 3^2 \times 5$, answer the following questions with reasons:

1. a) Is $N$ a multiple of $12$?
2. b) Is $15$ a factor of $N$?
3. c) How many factors does the number $N$ have in total? (Use the formula based on exponents).

The floor of a rectangular garage measuring $12.6$ metres long by $4.2$ metres wide is to be paved with square tiles that are as large as possible. The builder says he does not want to cut any tiles.

1. a) What should the dimensions of each tile be? (Express the result in centimetres).
2. b) How many tiles will be needed in total to cover the entire floor?

Two natural numbers $a$ and $b$ have a product of $a \times b = 1512$. It is known that their highest common factor is $m.c.d.(a, b) = 6$.

Using the property relating the product of two numbers to their $m.c.d.$ and their $m.c.m.$, calculate the lowest common multiple of those numbers.

In a logistics warehouse, the shelves are numbered from $1$ to $100$. A worker must place security labels following this pattern:

• Place an orange label on all shelves that are multiples of $4$.

• Place a white label on all shelves that are multiples of $6$.

How many shelves will have both labels at the same time? State the numbers of the first three shelves that will have both colours.

Two city bus lines in Cork coincide at the same stop at $7:30\text{ h}$ in the morning. The circular line passes by that stop every $15\text{ minutes}$ and the downtown line passes every $25\text{ minutes}$.

1. a) Before calculating the least common multiple, make a reasoned estimate: will they coincide again before a full hour has passed since their departure, or will it take more than an hour? Explain your reasoning by mentally comparing the first few multiples of both frequencies.
2. b) Calculate the $\text{l.c.m.}(15, 25)$ by listing the multiples of each frequency and determine at what exact time they will coincide again at that stop for the first time.
3. c) Check whether your initial estimate was correct and calculate how many times in total they will coincide at that stop between $7:30\text{ h}$ and $11:00\text{ h}$ in the morning (both times included).

To organize the books in the municipal library into boxes containing the same amount and without any left over, it is necessary to calculate the greatest common divisor of $36$ and $60$. Two students propose different strategies:

• Cian's method: He writes the complete list of all divisors of $36$, the list of all divisors of $60$, and selects the largest number that appears in both lists.

• Aoife's method: She decomposes both numbers into prime factors ($36 = 2^2 \times 3^2$ and $60 = 2^2 \times 3 \times 5$) and applies the rule of taking the common prime factors raised to the lowest exponent.

1. a) Apply both methods step by step to obtain the $\text{g.c.d.}(36, 60)$ and verify that you reach the same result.
2. b) Compare both approaches: which one is more intuitive for small numbers? If you had to find the g.c.d. of $1,260$ and $2,100$, which of the two methods would you choose? Justify your decision by explaining the drawbacks of the other method when working with large numbers.

At a cybersecurity agency, a programmer needs to create a five-digit passcode. The number, which we will call $N$, must simultaneously satisfy the following divisibility conditions:

• It is a palindromic number (it reads the same from left to right as from right to left).

• The units digit is the first even prime number.

• It is divisible by $3$, by $4$ and by $11$.

What is the value of this passcode $N$ if exactly three different digits appear in it?

A carpenter has three wooden battens measuring $180\text{ cm}$, $252\text{ cm}$ and $420\text{ cm}$. He wants to cut them into equal pieces of the maximum possible length to make mirror frames, so that no wood is left over from any batten.

1. How long should each piece be?

2. How many frames can he make if he needs exactly $4$ pieces for each frame?

Three elite athletes are training on a circular track. The first takes $84\text{ seconds}$ to complete one full lap, the second takes $96\text{ seconds}$ and the third $120\text{ seconds}$. If all three start together from the finish line at $9:00$ in the morning:

1. At what exact time will all three coincide at the finish line again for the first time?

2. How many laps will each of them have completed at that moment?

Find two natural numbers, $a$ and $b$, knowing that their highest common factor is $m.c.d.(a, b) = 15$ and that their product is $a \times b = 6750$. Note that neither of the two numbers can be $15$ itself. Is there more than one possible solution? Justify your answer.

A logistics warehouse is organising boxes of robotics kits. They have between $700$ and $900$ sensors. If they pack them in boxes of $12$, they have $7$ left over. If they pack them in boxes of $15$, they also have $7$ left over. And if they pack them in boxes of $18$, they have $7$ sensors left over again. How many sensors do they have in the warehouse exactly?

A natural number is called "special" if it can be written as $2^a \times 3^b$, where $a$ and $b$ are natural numbers that can be $0$, and it has exactly $10$ natural divisors.

Determine all the possible "special" numbers that satisfy this condition.

Two cargo ships leave the port of Cork. The ship Shannon leaves every $24$ days and the ship Liffey leaves every $35$ days. On the $1$ of January they coincided at the port.

1. How many times will they coincide at the port again over the course of a calendar year (consider $365$ days)?

2. If a third ship, the Lee, were to leave every certain number of days $x$, and it is known that the first time all three coincide is after $840$ days, what is the smallest possible value for $x$ (given $x > 1$)?

The floor of a rectangular room measuring $12.6\text{ metres}$ long by $7.2\text{ metres}$ wide is to be tiled with square tiles of the largest possible size, without having to cut any.

1. What should the side length of each tile be, expressed in centimetres?

2. If each tile costs $\text{€}\ 8.50$ and the tile adhesive required costs $\text{€}\ 2.25$ per square metre of surface area, what will be the total budget for the materials for the room's floor?

A rapeseed oil cooperative in Kilkenny has stored two tanks: one with $720\text{ liters}$ of the cold-pressed variety and another with $480\text{ liters}$ of the garlic-infused variety. They want to bottle all the oil, without mixing the two varieties, into identical containers with an integer capacity of liters, so that no oil is left over.

1. a) If there were no size limit and the goal were to use the smallest possible number of containers, what capacity should each one have, and what mathematical concept does it represent?
2. b) Due to ergonomics and Health and Safety Authority (HSA) manual handling guidelines, workers are prohibited from manually handling containers larger than $30\text{ liters}$. Knowing that the chosen capacity must be a common divisor of $720$ and $480$ (and therefore a divisor of their g.c.d.), determine the largest possible capacity per container that complies with the regulations.
3. c) With the capacity calculated in the previous section, how many containers will be filled with cold-pressed oil, how many with garlic-infused oil, and how many will be needed in total?

In a class discussion about divisibility, Cian makes the following conjecture: "If a natural number is divisible by $4$ and at the same time divisible by $6$, then it must necessarily be divisible by their product, that is, by $24$".

1. a) Find two numbers less than $50$ that are multiples of both $4$ and $6$ simultaneously, but are not divisible by $24$. What does the existence of these two counterexamples prove about Cian's statement?
2. b) Explain the conceptual error Cian is making using prime factor decomposition or the concept of the least common multiple: why does being a multiple of $4$ and $6$ not require being a multiple of $24$?
3. c) What condition must two natural numbers $a$ and $b$ satisfy for it to always be true that any common multiple of both is also a multiple of the product $a \times b$? Give an example of two numbers greater than $2$ that do satisfy that condition and justify why the rule works with them.