Divisibility, GCF, and LCM

On a streaming platform, series are grouped into seasons. If a series has a total of $42$ episodes, state which of the following statements are true and which are false, briefly reasoning your answer:

1. a) The number $6$ is a divisor of $42$.
2. b) $42$ is a multiple of $5$.
3. c) $42$ is divisible by $14$.

Work out mentally and write in your notebook:

1. a) The first five multiples of $12$.
2. b) All the divisors of the number $24$.
3. c) The multiples of $7$ between $30$ and $60$.

Look at the following polygonal numbers that represent the pentagonal numbers. The first terms are $1, 5, 12, 22...$

1. a) Draw in your notebook the arrangement of dots for the fifth pentagonal number.
2. b) Following the sequence, what is the numerical value of the fifth and sixth pentagonal numbers?
3. c) Check if the tenth pentagonal number ($145$) is divisible by $3$, by $5$ and by $11$.

Look at the numbers that appear on the following panel and classify them into prime numbers or composite numbers.

Find the prime factorization of the following natural numbers. Remember to write the final result as a product of powers:

1. a) $36$
2. b) $60$
3. c) $126$

Calculate the greatest common factor (GCF) and the least common multiple (LCM) of the following pairs of numbers:

1. a) $10$ and $15$
2. b) $20$ and $30$
3. c) $8$ and $12$

Knowing that the product of two numbers is $2160$ and that their greatest common factor (GCF) is $12$:

1. a) Calculate the least common multiple (LCM) of these numbers using the relationship between GCF and LCM.
2. b) If one of the numbers is $60$, what is the other number?
3. c) Find the prime factorization of both numbers to check that the results of the previous sections are correct.

Emma has $24$ star stickers and $16$ planet stickers. She wants to make identical gift envelopes (with the same number of stars and the same number of planets in each one) with no stickers left over.

What is the maximum number of envelopes she can prepare? How many stickers of each type will there be in each envelope?

An interior design company has two rolls of colored LED strips: a blue light one measuring $120$ feet and a warm light one measuring $156$ feet. They want to cut both rolls into pieces of the same length, as long as possible, with no material left over.

1. a) How long must each piece be?
2. b) How many pieces of each color will they get in total?

A logistics company must organize three types of aid packages into shipping containers. They have $144$ boxes of blankets, $120$ boxes of non-perishable food and $84$ boxes of hygiene kits.

They want to load the maximum number of identical containers, so that each container has the same amount of boxes of each type and no boxes are left over in the warehouse.

1. How many containers can be sent?

2. How many boxes of each type will there be in each container?

Two lights on a gaming keyboard flash differently. The blue light turns on every $6$ seconds and the orange light every $9$ seconds. If both have turned on at the same time right now, how many seconds will pass until they coincide turned on again for the first time?

Three streaming channels start broadcasting special content today at 10:00 in the morning.

• Channel A broadcasts a giveaway every $18$ hours.

• Channel B broadcasts an interview every $24$ hours.

• Channel C broadcasts a tournament every $30$ hours.

How many days must pass as a minimum for the three events to coincide again at the same time? At what time and on what day of the week will it happen if today is Monday?

On the main stage of a music festival, there are three robotic lighting systems. The first system emits a white flash every $18$ seconds, the second a blue flash every $24$ seconds and the third an orange flash every $30$ seconds.

At $21:00$ hours the three systems coincide for the first time. How many times will the three flashes coincide exactly again before the concert finishes at $22:25$ hours?

A town council wants to pave a rectangular plaza $45$ feet long by $24$ feet wide using square stone tiles. The architect wants the tiles to be as large as possible and that none have to be cut.

1. What must the side length of each tile be?

2. If for budget reasons they decide to use tiles half that size, how many tiles will they need now to cover the whole plaza?

The mechanism of an old clock uses two cogwheels that mesh together. Wheel $A$ has $54$ teeth and wheel $B$ has $36$ teeth. If we put a red mark on the point where the teeth touch right now:

1. How many teeth of wheel $A$ must pass for the marks to coincide again for the first time?

2. How many turns will each wheel have made at that moment?

Three artificial satellites orbit the Earth. The first completes an orbit every $90$ minutes, the second every $2$ hours and the third every $105$ minutes. If at $12:00$ noon they are aligned over the Greenwich meridian:

1. At what exact time will they be aligned again for the first time?

2. How many orbits will the slowest satellite have completed in that time?

A video game requires you to enter a four-digit code to unlock a level: $3.45\_$. Help the player find the last digit so that the number meets the following conditions:

1. a) The number is divisible by $2$ and by $5$ at the same time.
2. b) The number is divisible by $3$, but not by $2$.
3. c) It is a multiple of $9$.

A programmer is designing a four-digit access code. The code follows the format $35A2$, where $A$ is an unknown digit. For the code to be valid, the number must be divisible by $3$ and by $4$ simultaneously.

Determine all the possible values that the digit $A$ can take and write down the resulting complete numbers.

At a video game convention, they want to organize the participants into teams. We have a security code for access which is a four-digit number of the form $4A2B$.

For the code to be valid, it must meet two conditions:

1. a) It must be divisible by $3$.
2. b) It must be divisible by $11$.

Find all the possible values for the digits $A$ and $B$.

A famous influencer wants to protect access to his video editing account with a four-digit PIN code of the form $5X4Y$. So as not to forget it, he has memorized the following conditions based on divisibility:

• The number is divisible by $4$.

• The number is also divisible by $9$.

• The units digit ($Y$) is greater than the thousands digit.

Find all the possible PIN codes that meet these conditions.

Find a natural number that simultaneously meets the following conditions:

1. It is between $200$ and $250$.

2. It is a multiple of $13$.

3. When divided by $2$ and by $3$ it gives an exact division, but it is not divisible by $9$.

Calculate the total number of divisors that the number $360$ has without needing to write them all down. To do this:

1. a) Find its prime factorization using powers.
2. b) Apply the property of the product of the exponents increased by one.
3. c) State whether the number $360$ is a perfect square, reasoning your answer based on the exponents of its factorization.

A natural number $N$ has the following prime factorization: $N = 2^{a} \cdot 3^{2} \cdot 5^{1}$.

We know that the total number of divisors of $N$ is exactly $24$.

1. Calculate the value of the exponent $a$.

2. Determine the numerical value of $N$.

3. Write down all the divisors of $N$ that are multiples of $15$ but not of $2$.

It is known that a number $X$ is a multiple of $12$ and $15$ simultaneously. Furthermore, $X$ has exactly $10$ natural divisors.

1. Using prime factorization, deduce the value of $X$. (Hint: reason about the exponents of the factorization).

2. Justify whether the number $X+1$ is a prime or composite number.