Divisibility

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, giving a brief reason for your answer:

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

Look at the numbers appearing on the following board and classify them into prime numbers or composite numbers.

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 satisfies the following conditions:

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

Cian has to check on an exam whether the number $3,522$ is divisible by $6$. In his work, he writes the following: “Since the last digit is $2$, the number is divisible by $2$. However, since that last digit ($2$) is not divisible by $3$, the number $3,522$ cannot be divisible by $6$”.

1. a) Identify the conceptual error Cian makes when checking for divisibility by $3$ and explain why his reasoning is incorrect.
2. b) Correctly apply the divisibility rules for $2$ and $3$ to determine whether or not $3,522$ is divisible by $6$.

A municipal printing house in Kilkenny has produced $1,342$ calendars, and the workers must pack them into boxes of exactly $4$ units each, with none left over.

1. a) Without performing long division or applying the full rule yet, make a quick estimate: knowing that $1,300$ and $40$ are multiples of $4$, explain whether you think it is reasonable that some calendars will be left over.
2. b) Apply the divisibility rule for $4$ by looking exclusively at the last two digits to check your estimate.
3. c) In light of the result, state how many calendars are left over after filling the maximum number of full boxes of $4$, or how many calendars would be needed to complete one more box.

The public bike rental service in an Irish city identifies each vehicle using a four-digit code with the structure $4X6Y$, where $X$ and $Y$ are unknown digits.

For maintenance reasons:

• The bikes undergo a monthly workshop check-up if their code is divisible by $8$ (the last three digits form a multiple of $8$).

• In addition, they enter the special weekly disinfection shift if their code is divisible by $11$.

1. a) Apply the divisibility rule for $11$ to find what value the sum of the digits $X + Y$ must have.
2. b) Taking into account that to be divisible by $8$ the digit $Y$ must be even, analyze the possible pairs of digits $(X, Y)$ and find the only code $4X6Y$ that allows the bike to undergo both the monthly check-up and the weekly disinfection.

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 simultaneously divisible by $3$ and $4$.

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

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

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

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

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

A famous influencer wants to protect access to their video editing account with a four-digit PIN code of the form $5X4Y$. So as not to forget it, they have memorised 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 possible PIN codes that satisfy these conditions.

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

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

An interior design company has two rolls of coloured LED strips: one blue light strip measuring $120$ metres and one warm light strip measuring $156$ metres. They want to cut both rolls into pieces of equal length, as long as possible, leaving no material left over.

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

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 all three flashes coincide again before the concert ends at $22:25$ hours?

Calculate the total number of factors of the number $360$ without writing them all out. To do this:

1. a) Find its prime factorisation using powers.
2. b) Apply the rule of multiplying each exponent increased by one.
3. c) State whether the number $360$ is a perfect square, explaining your reasoning based on the exponents in its factorisation.

A natural number $N$ simultaneously satisfies the following three properties regarding its divisors and factors:

1. In its prime factorization, only $2$ and $3$ appear (both with an exponent greater than zero).

2. The total number of divisors of $N$ is exactly $9$.

3. $N$ is a perfect square.

1. a) Use the formula for the number of divisors and the property of the exponents of perfect squares to deduce, with justification, what the exponents of $2$ and $3$ must be in the factorization of $N$.
2. b) Reconstruct the number $N$, calculate its value, and write the complete list of its $9$ divisors to verify that it satisfies all the conditions.

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

We know that the total number of factors 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 factors of $N$ that are multiples of $15$ but not of $2$.