Multiples, Factors, GCF, and LCM

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

1. a) If they put $5$ games in each row, there will be $6$ rows. Then, $30$ is a ________ of $5$.
2. b) As the division $30 : 10$ is exact, we say that $10$ is a ________ of $30$.
3. c) The number $30$ is ________ by $2, 3$ and $5$ at the same time.

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

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

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

Apply the rules of divisibility to indicate whether the number $4.350$ is divisible by the following values (answer YES/NO and justify briefly):

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 $10$?

In an electronics store, they have received a batch of tablets. The manager knows that the total number of tablets is a four-digit number of the form $4A7B$. To organize the inventory, he needs the number to meet 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 the possible values of $A$ and $B$ so that the manager can organize his inventory.

In a cybersecurity agency, a programmer needs to create a five-digit access code. The number, which we will call $N$, must simultaneously meet the following divisibility conditions:

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

• The ones digit is the first even prime number.

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

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

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

Explain why you have chosen the number $21$ as composite by indicating which numbers it can be divided by in addition to $1$ and itself.

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

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

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

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

Natural numbers can be represented by rectangles according to their divisors. For example, the number $12$ can be represented as a rectangle of $3 \times 4$ or $2 \times 6$.

Look at the following drawing 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 the base $1 \times n$ also counts).

A natural number is called "special" if its prime factorization is of the form $2^a \cdot 3^b$ and it has exactly $10$ natural factors.

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

Emma has $24$ red stickers and $36$ blue stickers. She wants to make identical surprise envelopes for her friends, so 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 there be in each of those envelopes?

An e-sports 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 ones they have and without mixing colors inside each individual package, but 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?

The floor of a rectangular garage $12,6$ feet long by $4,2$ feet wide is to be paved with the largest possible square tiles. The builder says he does not want to cut any tiles.

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

A carpenter has three strips of precious wood measuring $180\text{ cm}$, $252\text{ cm}$ and $420\text{ cm}$. They want to cut them into equal pieces of the greatest possible length to make mirror frames, so that no wood is left over from any strip.

1. How long must each piece be?

2. How many frames will they be able to make if they need exactly $4$ pieces for each frame?

We want to tile the floor of a rectangular room $12,6\text{ meters}$ long and $7,2\text{ meters}$ wide with square tiles of the largest possible size, without having to cut any.

1. What must the side of each tile be, expressed in inches?

2. If each tile costs $\$8,50\text{ \$}$ and the necessary tile adhesive costs $\$2,25\text{ \$}$ for each square foot of surface, what will be the total budget for the materials for the room's floor?

Two alarm clocks ring at different intervals. The first rings every $15$ minutes and the second every $20$ minutes. If they have just coincided right now at $08:00$ in the morning:

1. a) In how many minutes will they ring 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 with 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 have coincided in uploading a video today, Monday, at $09:00$ a.m., what day and at what time will all three coincide again for the first time?

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 colors.

Three elite athletes are training on a circular track. The first takes $84\text{ seconds}$ to complete one 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 the three coincide again for the first time at the finish line?

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

Two cargo ships leave the port of Miami. The ship Levante leaves every $24$ days and the ship Poniente leaves every $35$ days. On January $1$ they coincided in the port.

1. How many times will they coincide again in the port throughout a calendar year (consider $365$ days)?

2. If a third ship, the Tramontana, 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 $x$ (where $x > 1$)?

A logistics warehouse is organizing 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 again have $7$ sensors left over. Exactly how many sensors do they have in the warehouse?

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

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

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