Multiples and Divisors

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?

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 $10$?

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.

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?

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.

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

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

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).

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).

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 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?

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?

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?

Marta 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 color and none are left over.

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

An esports club wants to organize 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 color, distributing all the t-shirts they have and without mixing colors within each individual package, while maximizing 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 color will be obtained?

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?

The floor of a rectangular garage measuring $12.6$ meters long by $4.2$ meters 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 centimeters).
2. b) How many tiles will be needed in total to cover the entire floor?

The floor of a rectangular room measuring $12.6\text{ meters}$ long by $7.2\text{ meters}$ 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 centimeters?

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

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?